Chapter 5 Resource Masters©Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 5 Resource Masters The Fast File Chapter Resource system allows you to conveniently

Chapter 5Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 5 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828008-7 Algebra 2Chapter 5 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02

Glenc oe/ M cGr aw-H ill

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 5-1Study Guide and Intervention . . . . . . . . 239–240Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 241Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Reading to Learn Mathematics . . . . . . . . . . 243Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 244









Chapter 5 AssessmentChapter 5 Test, Form 1 . . . . . . . . . . . . 293–294Chapter 5 Test, Form 2A . . . . . . . . . . . 295–296Chapter 5 Test, Form 2B . . . . . . . . . . . 297–298Chapter 5 Test, Form 2C . . . . . . . . . . . 299–300Chapter 5 Test, Form 2D . . . . . . . . . . . 301–302Chapter 5 Test, Form 3 . . . . . . . . . . . . 303–304Chapter 5 Open-Ended Assessment . . . . . . 305Chapter 5 Vocabulary Test/Review . . . . . . . 306Chapter 5 Quizzes 1 & 2 . . . . . . . . . . . . . . . 307Chapter 5 Quizzes 3 & 4 . . . . . . . . . . . . . . . 308Chapter 5 Mid-Chapter Test . . . . . . . . . . . . . 309Chapter 5 Cumulative Review . . . . . . . . . . . 310Chapter 5 Standardized Test Practice . .311–312

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A38

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 5 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 5 Resource Masters includes the core materials neededfor Chapter 5. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 5-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 5Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 282–283. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

55

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 5.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

binomial

coefficient

KOH·uh·FIH·shuhnt

complex conjugates

KAHN·jih·guht

complex number

degree

extraneous solution

ehk·STRAY·nee·uhs

FOIL method

imaginary unit

like radical expressions

like terms

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

monomial

nth root

polynomial

power

principal root

pure imaginary number

radical equation

radical inequality

rationalizing the denominator

synthetic division

sihn·THEH·tihk

trinomial

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

55

Study Guide and InterventionMonomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

© Glencoe/McGraw-Hill 239 Glencoe Algebra 2

Less

on

5-1

Monomials A monomial is a number, a variable, or the product of a number and one ormore variables. Constants are monomials that contain no variables.

Negative Exponent a!n " and " an for any real number a # 0 and any integer n.

When you simplify an expression, you rewrite it without parentheses or negativeexponents. The following properties are useful when simplifying expressions.

Product of Powers am $ an " am % n for any real number a and integers m and n.

Quotient of Powers " am ! n for any real number a # 0 and integers m and n.

For a, b real numbers and m, n integers:(am)n " amn

Properties of Powers(ab)m " ambm

! "n" , b # 0

! "!n" ! "n

or , a # 0, b # 0

Simplify. Assume that no variable equals 0.

bn&an

b&a

a&b

an&bn

a&b

am&an

1&a!n

1&an

ExampleExample

a. (3m4n!2)(!5mn)2

(3m4n!2)(!5mn)2 " 3m4n!2 $ 25m2n2

" 75m4m2n!2n2

" 75m4 % 2n!2 % 2

" 75m6

b.

"

" !m12 $ 4m4

" !4m16

!m12&&4m

14&

(!m4)3&(2m2)!2

(!m4)3""(2m2)!2

ExercisesExercises


1. c12 $ c!4 $ c6 c14 2. b6 3. (a4)5 a20

4. 5. ! "!16. ! "2

7. (!5a2b3)2(abc)2 5a6b8c2 8. m7 $ m8 m15 9.

10. 2 11. 4j(2j!2k2)(3j3k!7) 12. m23"2

2mn2(3m2n)2&&

12m3n424j2"

k523c4t2&22c4t2

2m2"n

8m3n2&4mn3

1&5

x2"y4

x2y&xy3

b"a5

a2b&a!3b2

y2"x6

x!2y&x4y!1

b8&b2


Scientific Notation

Scientific notation A number expressed in the form a ' 10n, where 1 ( a ) 10 and n is an integer

Express 46,000,000 in scientific notation.

46,000,000 " 4.6 ' 10,000,000 1 ( 4.6 ) 10" 4.6 ' 107 Write 10,000,000 as a power of ten.

Evaluate . Express the result in scientific notation.

" '

" 0.7 ' 106

" 7 ' 105

Express each number in scientific notation.

1. 24,300 2. 0.00099 3. 4,860,0002.43 # 104 9.9 # 10!4 4.86 # 106

4. 525,000,000 5. 0.0000038 6. 221,0005.25 # 108 3.8 # 10!6 2.21 # 105

7. 0.000000064 8. 16,750 9. 0.0003696.4 # 10!8 1.675 # 104 3.69 # 10!4

Evaluate. Express the result in scientific notation.

10. (3.6 ' 104)(5 ' 103) 11. (1.4 ' 10!8)(8 ' 1012) 12. (4.2 ' 10!3)(3 ' 10!2)1.8 # 108 1.12 # 105 1.26 # 10!4

13. 14. 15.

2.5 # 109 9 # 10!8 7.4 # 103

16. (3.2 ' 10!3)2 17. (4.5 ' 107)2 18. (6.8 ' 10!5)2

1.024 # 10!5 2.025 # 1015 4.624 # 10!9

19. ASTRONOMY Pluto is 3,674.5 million miles from the sun. Write this number inscientific notation. Source: New York Times Almanac 3.6745 # 109 miles

20. CHEMISTRY The boiling point of the metal tungsten is 10,220°F. Write thistemperature in scientific notation. Source: New York Times Almanac 1.022 # 104

21. BIOLOGY The human body contains 0.0004% iodine by weight. How many pounds ofiodine are there in a 120-pound teenager? Express your answer in scientific notation.Source: Universal Almanac 4.8 # 10!4 lb

4.81 ' 108&&6.5 ' 104

1.62 ' 10!2&&

1.8 ' 1059.5 ' 107&&3.8 ' 10!2

104&10!2

3.5&5

3.5 ' 104&&5 ' 10!2

3.5 ' 104""5 ' 10!2

Study Guide and Intervention (continued)

Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeMonomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1


Less

on

5-1


1. b4 $ b3 b7 2. c5 $ c2 $ c2 c9

3. a!4 $ a!3 4. x5 $ x!4 $ x x2

5. (g4)2 g8 6. (3u)3 27u3

7. (!x)4 x4 8. !5(2z)3 !40z3

9. !(!3d)4 !81d4 10. (!2t2)3 !8t6

11. (!r7)3 !r 21 12. s3

13. 14. (!3f 3g)3 !27f 9g3

15. (2x)2(4y)2 64x2y2 16. !2gh( g3h5) !2g4h6

17. 10x2y3(10xy8) 100x3y11 18.

19. ! 20.


21. 53,000 5.3 # 104 22. 0.000248 2.48 # 10!4

23. 410,100,000 4.101 # 108 24. 0.00000805 8.05 # 10!6


25. (4 ' 103)(1.6 ' 10!6) 6.4 # 10!3 26. 6.4 # 10109.6 ' 107&&1.5 ' 10!3

2q2"p2

!10pq4r&&!5p3q2r

c7"6a3b

!6a4bc8&&36a7b2c

8z2"w2

24wz7&3w3z5

1"k

k9&k10

s15&s12

1"a7



1. n5 $ n2 n7 2. y7 $ y3 $ y2 y12

3. t9 $ t!8 t 4. x!4 $ x!4 $ x4

5. (2f 4)6 64f 24 6. (!2b!2c3)3 !

7. (4d2t5v!4)(!5dt!3v!1) ! 8. 8u(2z)3 64uz3

9. ! 10. !

11. 12. ! "2

13. !(4w!3z!5)(8w)2 ! 14. (m4n6)4(m3n2p5)6 m34n36p30

15. ! d2f 4"4!! d5f"3 !12d23f 19 16. ! "!2

17. 18. !


19. 896,000 20. 0.000056 21. 433.7 ' 108

8.96 # 105 5.6 # 10!5 4.337 # 1010


22. (4.8 ' 102)(6.9 ' 104) 23. (3.7 ' 109)(8.7 ' 102) 24.

3.312 # 107 3.219 # 1012 3 # 10!5

25. COMPUTING The term bit, short for binary digit, was first used in 1946 by John Tukey.A single bit holds a zero or a one. Some computers use 32-bit numbers, or strings of 32 consecutive bits, to identify each address in their memories. Each 32-bit numbercorresponds to a number in our base-ten system. The largest 32-bit number is nearly4,295,000,000. Write this number in scientific notation. 4.295 # 109

26. LIGHT When light passes through water, its velocity is reduced by 25%. If the speed oflight in a vacuum is 1.86 ' 105 miles per second, at what velocity does it travel throughwater? Write your answer in scientific notation. 1.395 # 105 mi/s

27. TREES Deciduous and coniferous trees are hard to distinguish in a black-and-whitephoto. But because deciduous trees reflect infrared energy better than coniferous trees,the two types of trees are more distinguishable in an infrared photo. If an infraredwavelength measures about 8 ' 10!7 meters and a blue wavelength measures about 4.5 ' 10!7 meters, about how many times longer is the infrared wavelength than theblue wavelength? about 1.8 times

2.7 ' 106&&9 ' 1010

4v2"m2

!20(m2v)(!v)3&&

5(!v)2(!m4)15x11"y3

(3x!2y3)(5xy!8)&&

(x!3)4y!2

y6"4x2

2x3y2&!x2y5

4&3

3&2

256"wz5

4"9r4s6z12

2&3r2s3z6

27x6"16

!27x3(!x7)&&

16x4

s4"3x4

!6s5x3&18sx7

4m7y2"3

12m8y6&!9my4

20d 3t2"v5

8c9"b6

1"x4

Practice (Average)

Monomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

Reading to Learn MathematicsMonomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1


Less

on

5-1

Pre-Activity Why is scientific notation useful in economics?

Read the introduction to Lesson 5-1 at the top of page 222 in your textbook.

Your textbook gives the U.S. public debt as an example from economics thatinvolves large numbers that are difficult to work with when written instandard notation. Give an example from science that involves very largenumbers and one that involves very small numbers. Sample answer:distances between Earth and the stars, sizes of molecules and atoms

Reading the Lesson

1. Tell whether each expression is a monomial or not a monomial. If it is a monomial, tellwhether it is a constant or not a constant.

a. 3x2 monomial; not a constant b. y2 % 5y ! 6 not a monomial

c. !73 monomial; constant d. not a monomial

2. Complete the following definitions of a negative exponent and a zero exponent.

For any real number a # 0 and any integer n, a!n " .

For any real number a # 0, a0 " .

3. Name the property or properties of exponents that you would use to simplify eachexpression. (Do not actually simplify.)

a. quotient of powers

b. y6 $ y9 product of powers

c. (3r2s)4 power of a product and power of a power

Helping You Remember

4. When writing a number in scientific notation, some students have trouble rememberingwhen to use positive exponents and when to use negative ones. What is an easy way toremember this? Sample answer: Use a positive exponent if the number is10 or greater. Use a negative number if the number is less than 1.

m8&m3

1

"a1n"

1&z


Properties of ExponentsThe rules about powers and exponents are usually given with letters such as m, n,and k to represent exponents. For example, one rule states that am $ an " am % n.

In practice, such exponents are handled as algebraic expressions and the rules of algebra apply.

Simplify 2a2(an $ 1 $ a4n).

2a2(an % 1 % a4n) " 2a2 $ an % 1 % 2a2 $ a4n Use the Distributive Law.

" 2a2 % n % 1 % 2a2 % 4n Recall am $ an " am % n.

" 2an % 3 % 2a2 % 4n Simplify the exponent 2 % n % 1 as n % 3.

It is important always to collect like terms only.

Simplify (an $ bm)2.

(an % bm)2 " (an % bm)(an % bm)F O I L

" an $ an % an $ bm % an $ bm % bm $ bm The second and third terms are like terms.

" a2n % 2anbm % b2m

Simplify each expression by performing the indicated operations.

1. 232m 2. (a3)n 3. (4nb2)k

4. (x3aj )m 5. (!ayn)3 6. (!bkx)2

7. (c2)hk 8. (!2dn)5 9. (a2b)(anb2)

10. (xnym)(xmyn) 11. 12.

13. (ab2 ! a2b)(3an % 4bn)

14. ab2(2a2bn ! 1 % 4abn % 6bn % 1)

12x3&4xn

an&2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-15-1

Example 1Example 1

Example 2Example 2

Study Guide and InterventionPolynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2


Less

on

5-2

Add and Subtract Polynomials

Polynomial a monomial or a sum of monomials

Like Terms terms that have the same variable(s) raised to the same power(s)

To add or subtract polynomials, perform the indicated operations and combine like terms.

Simplify !6rs $ 18r2 ! 5s2 ! 14r2 $ 8rs ! 6s2.

!6rs % 18r2 ! 5s2 !14r2 % 8rs! 6s2

" (18r2 ! 14r2) % (!6rs % 8rs) % (!5s2 ! 6s2) Group like terms." 4r2 % 2rs ! 11s2 Combine like terms.

Simplify 4xy2 $ 12xy ! 7x2y ! (20xy $ 5xy2 ! 8x2y).

4xy2 % 12xy ! 7x2y ! (20xy % 5xy2 ! 8x2y)" 4xy2 % 12xy ! 7x2y ! 20xy ! 5xy2 % 8x2y Distribute the minus sign." (!7x2y % 8x2y ) % (4xy2 ! 5xy2) % (12xy ! 20xy) Group like terms." x2y ! xy2 ! 8xy Combine like terms.

Simplify.

1. (6x2 ! 3x % 2) ! (4x2 % x ! 3) 2. (7y2 % 12xy ! 5x2) % (6xy ! 4y2 ! 3x2)2x2 ! 4x $ 5 3y2$ 18xy ! 8x2

3. (!4m2 ! 6m) ! (6m % 4m2) 4. 27x2 ! 5y2 % 12y2 ! 14x2

!8m2 ! 12m 13x2 $ 7y2

5. (18p2 % 11pq ! 6q2) ! (15p2 ! 3pq % 4q2) 6. 17j2 ! 12k2 % 3j2 ! 15j2 % 14k2

3p2 $ 14pq ! 10q2 5j2 $ 2k2

7. (8m2 ! 7n2) ! (n2 ! 12m2) 8. 14bc % 6b ! 4c % 8b ! 8c % 8bc20m2 ! 8n2 14b $ 22bc ! 12c

9. 6r2s % 11rs2 % 3r2s ! 7rs2 % 15r2s ! 9rs2 10. !9xy % 11x2 ! 14y2 ! (6y2 ! 5xy ! 3x2)24r2s ! 5rs2 14x2 ! 4xy ! 20y2

11. (12xy ! 8x % 3y) % (15x ! 7y ! 8xy) 12. 10.8b2 ! 5.7b % 7.2 ! (2.9b2 ! 4.6b ! 3.1)7x $ 4xy ! 4y 7.9b2 ! 1.1b $ 10.3

13. (3bc ! 9b2 ! 6c2) % (4c2 ! b2 % 5bc) 14. 11x2 % 4y2 % 6xy % 3y2 ! 5xy ! 10x2

!10b2 $ 8bc ! 2c2 x2 $ xy $ 7y2

15. x2 ! xy % y2 ! xy % y2 ! x2 16. 24p3 ! 15p2 % 3p ! 15p3 % 13p2 ! 7p

! x2 ! xy $ y2 9p3 ! 2p2 ! 4p3"4

7"8

1"8

3&8

1&4

1&2

1&2

3&8

1&4

Example 1Example 1

Example 2Example 2

ExercisesExercises


Multiply Polynomials You use the distributive property when you multiplypolynomials. When multiplying binomials, the FOIL pattern is helpful.

To multiply two binomials, add the products ofF the first terms,

FOIL Pattern O the outer terms,I the inner terms, andL the last terms.

Find 4y(6 ! 2y $ 5y2).

4y(6 ! 2y % 5y2) " 4y(6) % 4y(!2y) % 4y(5y2) Distributive Property" 24y ! 8y2 % 20y3 Multiply the monomials.

Find (6x ! 5)(2x $ 1).

(6x ! 5)(2x % 1) " 6x $ 2x % 6x $ 1 % (!5) $ 2x % (!5) $ 1First terms Outer terms Inner terms Last terms

" 12x2 % 6x ! 10x ! 5 Multiply monomials." 12x2 ! 4x ! 5 Add like terms.

Find each product.

1. 2x(3x2 ! 5) 2. 7a(6 ! 2a ! a2) 3. !5y2( y2 % 2y ! 3)6x3 ! 10x 42a ! 14a2 ! 7a3 !5y4 ! 10y3 $ 15y2

4. (x ! 2)(x % 7) 5. (5 ! 4x)(3 ! 2x) 6. (2x ! 1)(3x % 5)x2 $ 5x ! 14 15 ! 22x $ 8x2 6x2 $ 7x ! 5

7. (4x % 3)(x % 8) 8. (7x ! 2)(2x ! 7) 9. (3x ! 2)(x % 10)4x2 $ 35x $ 24 14x2 ! 53x $ 14 3x2 $ 28x ! 20

10. 3(2a % 5c) ! 2(4a ! 6c) 11. 2(a ! 6)(2a % 7) 12. 2x(x % 5) ! x2(3 ! x)!2a $ 27c 4a2 ! 10a ! 84 x3 ! x2 $ 10x

13. (3t2 ! 8)(t2 % 5) 14. (2r % 7)2 15. (c % 7)(c ! 3)3t4 $ 7t2 ! 40 4r2 $ 28r $ 49 c2 $ 4c ! 21

16. (5a % 7)(5a ! 7) 17. (3x2 ! 1)(2x2 % 5x)25a2 ! 49 6x4 $ 15x3 ! 2x2 ! 5x

18. (x2 ! 2)(x2 ! 5) 19. (x % 1)(2x2 ! 3x % 1)x4 ! 7x2 $ 10 2x3 ! x2 ! 2x $ 1

20. (2n2 ! 3)(n2 % 5n ! 1) 21. (x ! 1)(x2 ! 3x % 4)2n4 $ 10n3 ! 5n2 ! 15n $ 3 x3 ! 4x2 $ 7x ! 4


Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticePolynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2


Less

on

5-2

Determine whether each expression is a polynomial. If it is a polynomial, state thedegree of the polynomial.

1. x2 % 2x % 2 yes; 2 2. no 3. 8xz % y yes; 2

Simplify.

4. (g % 5) % (2g % 7) 5. (5d % 5) ! (d % 1)3g $ 12 4d $ 4

6. (x2 ! 3x ! 3) % (2x2 % 7x ! 2) 7. (!2f 2 ! 3f ! 5) % (!2f 2 ! 3f % 8)3x2 $ 4x ! 5 !4f 2 ! 6f $ 3

8. (4r2 ! 6r % 2) ! (!r2 % 3r % 5) 9. (2x2 ! 3xy) ! (3x2 ! 6xy ! 4y2)5r2 ! 9r ! 3 !x2 $ 3xy $ 4y2

10. (5t ! 7) % (2t2 % 3t % 12) 11. (u ! 4) ! (6 % 3u2 ! 4u)2t2 $ 8t $ 5 !3u2 $ 5u ! 10

12. !5(2c2 ! d2) 13. x2(2x % 9)!10c2 $ 5d2 2x3 $ 9x2

14. 2q(3pq % 4q4) 15. 8w(hk2 % 10h3m4 ! 6k5w3)6pq2 $ 8q5 8hk2w $ 80h3m4w ! 48k5w4

16. m2n3(!4m2n2 ! 2mnp ! 7) 17. !3s2y(!2s4y2 % 3sy3 % 4)!4m4n5 ! 2m3n4p ! 7m2n3 6s6y3 ! 9s3y4 ! 12s2y

18. (c % 2)(c % 8) 19. (z ! 7)(z % 4)c2 $ 10c $ 16 z2 ! 3z ! 28

20. (a ! 5)2 21. (2x ! 3)(3x ! 5)a2 ! 10a $ 25 6x2 ! 19x $ 15

22. (r ! 2s)(r % 2s) 23. (3y % 4)(2y ! 3)r2 ! 4s2 6y2 ! y ! 12

24. (3 ! 2b)(3 % 2b) 25. (3w % 1)2

9 ! 4b2 9w2 $ 6w $ 1

1&2

b2c&d4


Determine whether each expression is a polynomial. If it is a polynomial, state thedegree of the polynomial.

1. 5x3 % 2xy4 % 6xy yes; 5 2. ! ac ! a5d3 yes; 8 3. no

4. 25x3z ! x#78$ yes; 4 5. 6c!2 % c ! 1 no 6. % no

Simplify.

7. (3n2 % 1) % (8n2 ! 8) 8. (6w ! 11w2) ! (4 % 7w2)11n2 ! 7 !18w2 $ 6w ! 4

9. (!6n ! 13n2) % (!3n % 9n2) 10. (8x2 ! 3x) ! (4x2 % 5x ! 3)!9n ! 4n2 4x2 ! 8x $ 3

11. (5m2 ! 2mp ! 6p2) ! (!3m2 % 5mp % p2) 12. (2x2 ! xy % y2) % (!3x2 % 4xy % 3y2)8m2 ! 7mp ! 7p2 !x2 $ 3xy $ 4y2

13. (5t ! 7) % (2t2 % 3t % 12) 14. (u ! 4) ! (6 % 3u2 ! 4u)2t2 $ 8t $ 5 !3u2 $ 5u ! 10

15. !9( y2 ! 7w) 16. !9r4y2(!3ry7 % 2r3y4 ! 8r10)!9y2 $ 63w 27r 5y9 ! 18r7y6 $ 72r14y2

17. !6a2w(a3w ! aw4) 18. 5a2w3(a2w6 ! 3a4w2 % 9aw6)!6a5w2 $ 6a3w5 5a4w9 ! 15a6w5 $ 45a3w9

19. 2x2(x2 % xy ! 2y2) 20. ! ab3d2(!5ab2d5 ! 5ab)

2x4 $ 2x3y ! 4x2y2 3a2b5d7 $ 3a2b4d 2

21. (v2 ! 6)(v2 % 4) 22. (7a % 9y)(2a ! y)v4 ! 2v2 ! 24 14a2 $ 11ay ! 9y2

23. ( y ! 8)2 24. (x2 % 5y)2

y2 ! 16y $ 64 x4 $ 10x2y $ 25y2

25. (5x % 4w)(5x ! 4w) 26. (2n4 ! 3)(2n4 % 3)25x2 ! 16w2 4n8 ! 9

27. (w % 2s)(w2 ! 2ws % 4s2) 28. (x % y)(x2 ! 3xy % 2y2)w3 $ 8s3 x3 ! 2x2y ! xy2 $ 2y3

29. BANKING Terry invests $1500 in two mutual funds. The first year, one fund grows 3.8%and the other grows 6%. Write a polynomial to represent the amount Terry’s $1500grows to in that year if x represents the amount he invested in the fund with the lessergrowth rate. !0.022x $ 1590

30. GEOMETRY The area of the base of a rectangular box measures 2x2 % 4x ! 3 squareunits. The height of the box measures x units. Find a polynomial expression for thevolume of the box. 2x3 $ 4x2 ! 3x units3

3&5

6&s

5&r

12m8n9&&(m ! n)2

4&3

Practice (Average)

Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

Reading to Learn MathematicsPolynomials

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5-25-2


Less

on

5-2

Pre-Activity How can polynomials be applied to financial situations?


Suppose that Shenequa decides to enroll in a five-year engineering programrather than a four-year program. Using the model given in your textbook,how could she estimate the tuition for the fifth year of her program? (Donot actually calculate, but describe the calculation that would be necessary.)Multiply $15,604 by 1.04.

Reading the Lesson

1. State whether the terms in each of the following pairs are like terms or unlike terms.

a. 3x2, 3y2 unlike terms b. !m4, 5m4 like termsc. 8r3, 8s3 unlike terms d. !6, 6 like terms

2. State whether each of the following expressions is a monomial, binomial, trinomial, ornot a polynomial. If the expression is a polynomial, give its degree.

a. 4r4 ! 2r % 1 trinomial; degree 4 b. #3x$ not a polynomialc. 5x % 4y binomial; degree 1 d. 2ab % 4ab2 ! 6ab3 trinomial; degree 4

3. a. What is the FOIL method used for in algebra? to multiply binomialsb. The FOIL method is an application of what property of real numbers?

Distributive Propertyc. In the FOIL method, what do the letters F, O, I, and L mean?

first, outer, inner, lastd. Suppose you want to use the FOIL method to multiply (2x % 3)(4x % 1). Show the

terms you would multiply, but do not actually multiply them.

F (2x)(4x)O (2x)(1)I (3)(4x)L (3)(1)


4. You can remember the difference between monomials, binomials, and trinomials bythinking of common English words that begin with the same prefixes. Give two wordsunrelated to mathematics that start with mono-, two that begin with bi-, and two thatbegin with tri-. Sample answer: monotonous, monogram; bicycle, bifocal;tricycle, tripod


Polynomials with Fractional CoefficientsPolynomials may have fractional coefficients as long as there are no variables in the denominators. Computing with fractional coefficients is performed in the same way as computing with whole-number coefficients.

Simpliply. Write all coefficients as fractions.

1. !&35&m ! &

27&p ! &

13&n" ! !&

73&p ! &

52&m ! &

34&n"

2.!&32&x ! &

43&y ! &

54&z" % !!&

14&x % y % &

25&z" % !!&

78&x ! &

67&y % &

12&z" "

38"

3.!&12&a2 ! &

13&ab % &

14&b2" % !&

56&a2 % &

23&ab ! &

34&b2" "

43

4.!&12&a2 ! &

13&ab % &

14&b2" ! !&

13&a2 ! &

12&ab % &

56&b2" "

5.!&12&a2 ! &

13&ab % &

14&b2" $ !&

12&a ! &

23&b" "

14"

6.!&23&a2 ! &

15&a % &

27&" $ !&

23&a3 % &

15&a2 ! &

27&a" "

7.!&23&x2 ! &

34&x ! 2" $ !&

45&x ! &

16&x2 ! &

12&"

8.!&16& % &

13&x % &

16&x4 ! &

12&x2" $ !&

16&x3 ! &

13& ! &

13&x" "3

1

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-25-2

Study Guide and InterventionDividing Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


Less

on

5-3

Use Long Division To divide a polynomial by a monomial, use the properties of powersfrom Lesson 5-1.

To divide a polynomial by a polynomial, use a long division pattern. Remember that onlylike terms can be added or subtracted.

Simplify .

" ! !

" p3!2t2!1r1!1 ! p2!2qt1!1r2!1 ! p3!2t1!1r1!1

" 4pt !7qr ! 3p

Use long division to find (x3 ! 8x2 $ 4x ! 9) % (x ! 4).

x2 ! 4x ! 12x ! 4%x$3$!$ 8$x$2$%$ 4$x$ !$ 9$

(!)x3 ! 4x2

!4x2 % 4x(!)!4x2 % 16x

!12x ! 9(!)!12x % 48

!57The quotient is x2 ! 4x ! 12, and the remainder is !57.

Therefore " x2 ! 4x ! 12 ! .

Simplify.

1. 2. 3.

6a2 $ 10a ! 10 5ab ! $ 7a4

4. (2x2 ! 5x ! 3) * (x ! 3) 5. (m2 ! 3m ! 7) * (m % 2)

2x $ 1 m ! 5 $

6. (p3 ! 6) * (p ! 1) 7. (t3 ! 6t2 % 1) * (t % 2)

p2 $ p $ 1 ! t2 ! 8t $ 16 !

8. (x5 ! 1) * (x ! 1) 9. (2x3 ! 5x2 % 4x ! 4) * (x ! 2)

x4 $ x3 $ x2 $ x $ 1 2x2 ! x $ 2

31"t $ 2

5"p ! 1

3"m $ 2

4b2"a

6n3"m

60a2b3 ! 48b4 % 84a5b2&&&

12ab224mn6 ! 40m2n3&&&

4m2n318a3 % 30a2&&3a

57&x ! 4

x3 ! 8x2 % 4x ! 9&&&x ! 4

9&3

21&3

12&3

9p3tr&3p2tr

21p2qtr2&&

3p2tr12p3t2r&

3p2tr12p3t2r ! 21p2qtr2 ! 9p3tr&&&&

3p2tr

12p3t2r ! 21p2qtr2 ! 9p3tr""""

3p2trExample 1Example 1

Example 2Example 2

ExercisesExercises

EVENS ONLY


Use Synthetic Division

Synthetic division a procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x ! r

Use synthetic division to find (2x3 ! 5x2 % 5x ! 2) * (x ! 1).

Step 1 Write the terms of the dividend so that the degrees of the terms are in 2x3 ! 5x2 % 5x ! 2 descending order. Then write just the coefficients. 2 !5 5 !2

Step 2 Write the constant r of the divisor x ! r to the left, In this case, r " 1. 1 2 !5 5 !2Bring down the first coefficient, 2, as shown.

2

Step 3 Multiply the first coefficient by r, 1 $ 2 " 2. Write their product under the 1 2 !5 5 !2 second coefficient. Then add the product and the second coefficient: 2 !5 % 2 " ! 3. 2 !3

Step 4 Multiply the sum, !3, by r : !3 $ 1 " !3. Write the product under the next 1 2 !5 5 !2 coefficient and add: 5 % (!3) " 2. 2 !3

2 !3 2

Step 5 Multiply the sum, 2, by r : 2 $ 1 " 2. Write the product under the next 1 2 !5 5 !2 coefficient and add: !2 % 2 " 0. The remainder is 0. 2 !3 2

2 !3 2 0

Thus, (2x3 ! 5x2 % 5x ! 2) * (x ! 1) " 2x2 ! 3x % 2.

Simplify.

1. (3x3 ! 7x2 % 9x ! 14) * (x ! 2) 2. (5x3 % 7x2 ! x ! 3) * (x % 1)

3x2 ! x $ 7 5x2 $ 2x ! 3

3. (2x3 % 3x2 ! 10x ! 3) * (x % 3) 4. (x3 ! 8x2 % 19x ! 9) * (x ! 4)

2x2 ! 3x ! 1 x2 ! 4x $ 3 $

5. (2x3 % 10x2 % 9x % 38) * (x % 5) 6. (3x3 ! 8x2 % 16x ! 1) * (x ! 1)

2x2 $ 9 ! 3x2 ! 5x $ 11 $

7. (x3 ! 9x2 % 17x ! 1) * (x ! 2) 8. (4x3 ! 25x2 % 4x % 20) * (x ! 6)

x2 ! 7x $ 3 $ 4x2 ! x ! 2 $

9. (6x3 % 28x2 ! 7x % 9) * (x % 5) 10. (x4 ! 4x3 % x2 % 7x ! 2) * (x ! 2)

6x2 ! 2x $ 3 ! x3 ! 2x2 ! 3x $ 1

11. (12x4 % 20x3 ! 24x2 % 20x % 35) * (3x % 5) 4x3 ! 8x $ 20 $ !65"3x $ 5

6"x $ 5

8"x ! 6

5"x ! 2

10"x ! 1

7"x $ 5

3"x ! 4


Dividing Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3

ExercisesExercises

EVENS ONLY

Skills PracticeDividing Polynomials

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Less

on

5-3

Simplify.

1. 5c $ 3 2. 3x $ 5

3. 5y2 $ 2y $ 1 4. 3x ! 1 !

5. (15q6 % 5q2)(5q4)!1 6. (4f 5 ! 6f4 % 12f 3 ! 8f 2)(4f 2)!1

3q2 $ f 3 ! $ 3f ! 2

7. (6j2k ! 9jk2) * 3jk 8. (4a2h2 ! 8a3h % 3a4) * (2a2)

2j ! 3k 2h2 ! 4ah $

9. (n2 % 7n % 10) * (n % 5) 10. (d2 % 4d % 3) * (d % 1)

n $ 2 d $ 3

11. (2s2 % 13s % 15) * (s % 5) 12. (6y2 % y ! 2)(2y ! 1)!1

2s $ 3 3y $ 2

13. (4g2 ! 9) * (2g % 3) 14. (2x2 ! 5x ! 4) * (x ! 3)

2g ! 3 2x $ 1 !

15. 16.

u $ 8 $ 2x $ 1 !

17. (3v2 ! 7v ! 10)(v ! 4)!1 18. (3t4 % 4t3 ! 32t2 ! 5t ! 20)(t % 4)!1

3v $ 5 $ 3t3 ! 8t2 ! 5

19. 20.

y2 ! 3y $ 6 ! 2x2 $ 5x ! 4 $

21. (4p3 ! 3p2 % 2p) * ( p ! 1) 22. (3c4 % 6c3 ! 2c % 4)(c % 2)!1

4p2 $ p $ 3 $ 3c3 ! 2 $

23. GEOMETRY The area of a rectangle is x3 % 8x2 % 13x ! 12 square units. The width ofthe rectangle is x % 4 units. What is the length of the rectangle? x2 $ 4x ! 3 units

8"c $ 2

3"p ! 1

3"x ! 3

18"y $ 2

2x3 ! x2 ! 19x % 15&&&x ! 3

y3 ! y2 ! 6&&y % 2

10"v ! 4

1"x ! 3

12"u ! 3

2x2 ! 5x ! 4&&x ! 3

u2 % 5u ! 12&&u ! 3

1"x ! 3

3a2"2

3f 2"2

1"q2

2"x

12x2 ! 4x ! 8&&4x

15y3 % 6y2 % 3y&&3y

12x % 20&&4

10c % 6&2


Simplify.

1. 3r6 ! r4 $ 2. ! 6k2 $

3. (!30x3y % 12x2y2 ! 18x2y) * (!6x2y) 4. (!6w3z4 ! 3w2z5 % 4w % 5z) * (2w2z)

5x ! 2y $ 3 !3wz3 ! $ $

5. (4a3 ! 8a2 % a2)(4a)!1 6. (28d3k2 % d2k2 ! 4dk2)(4dk2)!1

a2 ! 2a $ "a4" 7d2 $ "

d4" ! 1

7. f $ 5 8. 2x $ 7

9. (a3 ! 64) * (a ! 4) a2 $ 4a $ 16 10. (b3 % 27) * (b % 3) b2 ! 3b $ 9

11. 2x2 ! 8x $ 38 12. 2x2 ! 6x $ 22 !

13. (3w3 % 7w2 ! 4w % 3) * (w % 3) 14. (6y4 % 15y3 ! 28y ! 6) * (y % 2)

3w2 ! 2w $ 2 ! "w $3

3" 6y3 $ 3y2 ! 6y ! 16 $ "y2$6

2"

15. (x4 ! 3x3 ! 11x2 % 3x % 10) * (x ! 5) 16. (3m5 % m ! 1) * (m % 1)

x3 $ 2x2 ! x ! 2 3m4 ! 3m3 $ 3m2 ! 3m $ 4 !

17. (x4 ! 3x3 % 5x ! 6)(x % 2)!1 18. (6y2 ! 5y ! 15)(2y % 3)!1

x3 ! 5x2 $ 10x ! 15 $ "x2$4

2" 3y ! 7 $ "2y6$ 3"

19. 20.

2x $ 2 $ "2x1!2

3" 2x ! 1 ! "3x6$ 1"

21. (2r3 % 5r2 ! 2r ! 15) * (2r ! 3) 22. (6t3 % 5t2 ! 2t % 1) * (3t % 1)r2 $ 4r $ 5 2t2 $ t ! 1 $ "3t

2$ 1"

23. 24.

2p3 $ 3p2 ! 4p $ 1 2h2 ! h $ 325. GEOMETRY The area of a rectangle is 2x2 ! 11x % 15 square feet. The length of the

rectangle is 2x ! 5 feet. What is the width of the rectangle? x ! 3 ft

26. GEOMETRY The area of a triangle is 15x4 % 3x3 % 4x2 ! x ! 3 square meters. Thelength of the base of the triangle is 6x2 ! 2 meters. What is the height of the triangle?5x2 $ x $ 3 m

2h4 ! h3 % h2 % h ! 3&&&

h2 ! 14p4 ! 17p2 % 14p ! 3&&&2p ! 3

6x2 ! x ! 7&&3x % 1

4x2 ! 2x % 6&&2x ! 3

5"m $ 1

72"x $ 3

2x3 % 4x ! 6&&x % 3

2x3 % 6x % 152&&x % 4

2x2 % 3x ! 14&&x ! 2

f2 % 7f % 10&&f % 2

5"2w2

2"wz

3z4"2

9m"2k

3k"m

6k2m ! 12k3m2 % 9m3&&&

2km28"r2

15r10 ! 5r8 % 40r2&&&

5r4

Practice (Average)

Dividing Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3

Reading to Learn MathematicsDividing Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-35-3


Less

on

5-3

Pre-Activity How can you use division of polynomials in manufacturing?


Using the division symbol (*), write the division problem that you woulduse to answer the question asked in the introduction. (Do not actuallydivide.) (32x2 $ x) % (8x)

Reading the Lesson

1. a. Explain in words how to divide a polynomial by a monomial. Divide each term ofthe polynomial by the monomial.

b. If you divide a trinomial by a monomial and get a polynomial, what kind ofpolynomial will the quotient be? trinomial

2. Look at the following division example that uses the division algorithm for polynomials.

2x % 4x ! 4%$$2x2 ! 4x % 7

2x2 ! 8x4x % 74x ! 16

23Which of the following is the correct way to write the quotient? CA. 2x % 4 B. x ! 4 C. 2x % 4 % D.

3. If you use synthetic division to divide x3 % 3x2 ! 5x ! 8 by x ! 2, the division will looklike this:

2 1 3 !5 !82 10 10

1 5 5 2

Which of the following is the answer for this division problem? BA. x2 % 5x % 5 B. x2 % 5x % 5 %

C. x3 % 5x2 % 5x % D. x3 % 5x2 % 5x % 2


4. When you translate the numbers in the last row of a synthetic division into the quotientand remainder, what is an easy way to remember which exponents to use in writing theterms of the quotient? Sample answer: Start with the power that is one lessthan the degree of the dividend. Decrease the power by one for eachterm after the first. The final number will be the remainder. Drop any termthat is represented by a 0.

2&x ! 2

2&x ! 2

23&x ! 4

23&x ! 4


Oblique AsymptotesThe graph of y " ax % b, where a # 0, is called an oblique asymptote of y " f(x) if the graph of f comes closer and closer to the line as x → ∞ or x → !∞. ∞ is themathematical symbol for infinity, which means endless.

For f(x) " 3x % 4 % &2x&, y " 3x % 4 is an oblique asymptote because

f(x) ! 3x ! 4 " &2x&, and &

2x& → 0 as x → ∞ or !∞. In other words, as | x |

increases, the value of &2x& gets smaller and smaller approaching 0.

Find the oblique asymptote for f(x) & "x2 $

x8$x

2$ 15

".

!2 1 8 15 Use synthetic division.!2 !12

1 6 3

y " &x2 %

x8%x

2% 15& " x % 6 % &x %

32&

As | x | increases, the value of &x %3

2& gets smaller. In other words, since

&x %3

2& → 0 as x → ∞ or x → !∞, y " x % 6 is an oblique asymptote.

Use synthetic division to find the oblique asymptote for each function.

1. y " &8x2 !

x %4x

5% 11

&

2. y " &x2 %

x3!x

2! 15&

3. y " &x2 !

x2!x

3! 18&

4. y " &ax2

x%

!bx

d% c

&

5. y " &ax2

x%

%bx

d% c

&

Enrichment

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ExampleExample

Study Guide and InterventionFactoring Polynomials

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Less

on

5-4

Factor Polynomials

For any number of terms, check for:greatest common factor

For two terms, check for:Difference of two squares

a2 ! b2 " (a % b)(a ! b)Sum of two cubes

a3 % b3 " (a % b)(a2 ! ab % b2)Difference of two cubes

a3 ! b3 " (a ! b)(a2 % ab % b2)Techniques for Factoring Polynomials For three terms, check for:

Perfect square trinomialsa2 % 2ab % b2 " (a % b)2

a2 ! 2ab % b2 " (a ! b)2

General trinomialsacx2 % (ad % bc)x % bd " (ax % b)(cx % d )

For four terms, check for:Groupingax % bx % ay % by " x(a % b) % y(a % b)

" (a % b)(x % y)

Factor 24x2 ! 42x ! 45.

First factor out the GCF to get 24x2 ! 42x ! 45 " 3(8x2 ! 14x ! 15). To find the coefficientsof the x terms, you must find two numbers whose product is 8 $ (!15) " !120 and whosesum is !14. The two coefficients must be !20 and 6. Rewrite the expression using !20x and6x and factor by grouping.

8x2 ! 14x ! 15 " 8x2 ! 20x % 6x ! 15 Group to find a GCF." 4x(2x ! 5) % 3(2x ! 5) Factor the GCF of each binomial." (4x % 3)(2x ! 5) Distributive Property

Thus, 24x2 ! 42x ! 45 " 3(4x % 3)(2x ! 5).

Factor completely. If the polynomial is not factorable, write prime.

1. 14x2y2 % 42xy3 2. 6mn % 18m ! n ! 3 3. 2x2 % 18x % 16

14xy2(x $ 3y) (6m ! 1)(n $ 3) 2(x $ 8)(x $ 1)

4. x4 ! 1 5. 35x3y4 ! 60x4y 6. 2r3 % 250

(x2 $ 1)(x $ 1)(x ! 1) 5x3y(7y3 ! 12x) 2(r $ 5)(r2 ! 5r $ 25)

7. 100m8 ! 9 8. x2 % x % 1 9. c4 % c3 ! c2 ! c

(10m4 ! 3)(10m4 $ 3) prime c(c $ 1)2(c ! 1)

ExampleExample

ExercisesExercises

EVENS ONLY


Simplify Quotients In the last lesson you learned how to simplify the quotient of twopolynomials by using long division or synthetic division. Some quotients can be simplified byusing factoring.

Simplify .

" Factor the numerator and denominator.

" Divide. Assume x # 8, ! .

Simplify. Assume that no denominator is equal to 0.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

9x2 $ 6x $ 4""3x $ 2

y2 $ 4y $ 16""3y ! 5

2x ! 3""4x2 ! 2x $ 1

27x3 ! 8&&9x2 ! 4

y3 ! 64&&3y2 ! 17y % 20

4x2 ! 4x ! 3&&8x3 % 1

2x ! 3"x $ 8

7x ! 3"x ! 2

x $ 9"2x ! 5

4x2 ! 12x % 9&&2x2 % 13x ! 24

7x2 % 11x ! 6&&x2 ! 4

x2 ! 81&&2x2 ! 23x % 45

x ! 7"x $ 5

3x ! 5"2x ! 3

2x $ 5"x ! 1

x2 ! 14x % 49&&x2 ! 2x ! 35

3x2 % 4x ! 15&&2x2 % 3x ! 9

4x2 % 16x % 15&&2x2 % x ! 3

x ! 2"3x $ 7

6x $ 1"x $ 2

2x ! 1"x ! 2

x2 ! 7x % 10&&3x2 ! 8x ! 35

6x2 % 25x % 4&&x2 % 6x % 8

4x2 % 4x ! 3&&2x2 ! x ! 6

5x ! 1"x $ 3

2x ! 1"4x ! 1

x $ 3"x ! 5

5x2 % 9x ! 2&&x2 % 5x % 6

2x2 % 5x ! 3&&4x2 % 11x ! 3

x2 % x ! 6&&x2 ! 7x % 10

x ! 5"x $ 1

x $ 5"2x! 3

x ! 4"x $ 2

x2 ! 11x % 30&&x2 ! 5x ! 6

x2 % 6x % 5&&2x2 ! x ! 3

x2 ! 7x % 12&&x2 ! x ! 6

3&2

x % 4&x ! 8

(2x % 3)( x % 4)&&(x ! 8)(2x % 3)

8x2 % 11x % 12&&2x2 ! 13x ! 24

8x2 % 11x % 12""2x2 ! 13x ! 24


Factoring Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

ExampleExample

ExercisesExercises

1st COLUMN———#1, 4, 7, 10, 13

Skills PracticeFactoring Polynomials

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Less

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1. 7x2 ! 14x 2. 19x3 ! 38x2

7x(x ! 2) 19x2(x ! 2)

3. 21x3 ! 18x2y % 24xy2 4. 8j3k ! 4jk3 ! 73x(7x2 ! 6xy $ 8y2) prime

5. a2 % 7a ! 18 6. 2ak ! 6a % k ! 3(a $ 9)(a ! 2) (2a $ 1)(k ! 3)

7. b2 % 8b % 7 8. z2 ! 8z ! 10(b $ 7)(b $ 1) prime

9. m2 % 7m ! 18 10. 2x2 ! 3x ! 5(m ! 2)(m $ 9) (2x ! 5)(x $ 1)

11. 4z2 % 4z ! 15 12. 4p2 % 4p ! 24(2z $ 5)(2z ! 3) 4(p ! 2)(p $ 3)

13. 3y2 % 21y % 36 14. c2 ! 1003(y $ 4)(y $ 3) (c $ 10)(c ! 10)

15. 4f2 ! 64 16. d2 ! 12d % 364(f $ 4)(f ! 4) (d ! 6)2

17. 9x2 % 25 18. y2 % 18y % 81prime (y $ 9)2

19. n3 ! 125 20. m4 ! 1(n ! 5)(n2 $ 5n $ 25) (m2 $ 1)(m ! 1)(m $ 1)


21. 22.

23. 24.x ! 1"x ! 7

x2 % 6x ! 7&&

x2 ! 49x ! 5"x

x2 ! 10x % 25&&

x2 ! 5x

x $ 1"x $ 3

x2 % 4x % 3&&x2 % 6x % 9

x ! 2"x ! 5

x2 % 7x ! 18&&x2 % 4x ! 45



1. 15a2b ! 10ab2 2. 3st2 ! 9s3t % 6s2t2 3. 3x3y2 ! 2x2y % 5xy5ab(3a ! 2b) 3st(t ! 3s2 $ 2st) xy(3x2y ! 2x $ 5)

4. 2x3y ! x2y % 5xy2 % xy3 5. 21 ! 7t % 3r ! rt 6. x2 ! xy % 2x ! 2yxy(2x2 ! x $ 5y $ y2) (7 $ r )(3 ! t) (x $ 2)(x ! y)

7. y2 % 20y % 96 8. 4ab % 2a % 6b % 3 9. 6n2 ! 11n ! 2(y $ 8)(y $ 12) (2a $ 3)(2b $ 1) (6n $ 1)(n ! 2)

10. 6x2 % 7x ! 3 11. x2 ! 8x ! 8 12. 6p2 ! 17p ! 45(3x ! 1)(2x $ 3) prime (2p ! 9)(3p $ 5)

13. r3 % 3r2 ! 54r 14. 8a2 % 2a ! 6 15. c2 ! 49r (r $ 9)(r ! 6) 2(4a ! 3)(a $ 1) (c ! 7)(c $ 7)

16. x3 % 8 17. 16r2 ! 169 18. b4 ! 81(x $ 2)(x2 ! 2x $ 4) (4r $ 13)(4r ! 13) (b2 $ 9)(b $ 3)(b ! 3)

19. 8m3 ! 25 prime 20. 2t3 % 32t2 % 128t 2t(t $ 8)2

21. 5y5 % 135y2 5y2(y $ 3)(y2 ! 3y $ 9) 22. 81x4 ! 16 (9x2 $ 4)(3x $ 2)(3x ! 2)


23. 24. 25.

26. DESIGN Bobbi Jo is using a software package to create a drawing of a cross section of a brace as shown at the right.Write a simplified, factored expression that represents the area of the cross section of the brace. x(20.2 ! x) cm2

27. COMBUSTION ENGINES In an internal combustion engine, the up and down motion of the pistons is converted into the rotary motion of the crankshaft, which drives the flywheel. Let r1 represent the radius of the flywheel at the right and let r2 represent the radius of thecrankshaft passing through it. If the formula for the area of a circle is A " +r2, write a simplified, factored expression for the area of the cross section of the flywheel outside the crankshaft. ' (r1 ! r2)(r1 $ r2)

r1r2

x cm

x cm

8.2 cm

12 c

m

3(x $ 3)""x2 $ 3x $ 9

3x2 ! 27&&x3 ! 27

x ! 8"x $ 9

x2 ! 16x % 64&&

x2 % x ! 72x $ 4"x $ 5

x2 ! 16&&x2 % x ! 20

Practice (Average)

Factoring Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

Reading to Learn MathematicsFactoring Polynomials

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity How does factoring apply to geometry?


If a trinomial that represents the area of a rectangle is factored into twobinomials, what might the two binomials represent? the length andwidth of the rectangle

Reading the Lesson

1. Name three types of binomials that it is always possible to factor. difference of twosquares, sum of two cubes, difference of two cubes

2. Name a type of trinomial that it is always possible to factor. perfect squaretrinomial

3. Complete: Since x2 % y2 cannot be factored, it is an example of a polynomial.

4. On an algebra quiz, Marlene needed to factor 2x2 ! 4x ! 70. She wrote the followinganswer: (x % 5)(2x ! 14). When she got her quiz back, Marlene found that she did notget full credit for her answer. She thought she should have gotten full credit because shechecked her work by multiplication and showed that (x % 5)(2x ! 14) " 2x2 ! 4x ! 70.

a. If you were Marlene’s teacher, how would you explain to her that her answer was notentirely correct? Sample answer: When you are asked to factor apolynomial, you must factor it completely. The factorization was notcomplete, because 2x ! 14 can be factored further as 2(x ! 7).

b. What advice could Marlene’s teacher give her to avoid making the same kind of errorin factoring in the future? Sample answer: Always look for a commonfactor first. If there is a common factor, factor it out first, and then seeif you can factor further.


5. Some students have trouble remembering the correct signs in the formulas for the sumand difference of two cubes. What is an easy way to remember the correct signs?Sample answer: In the binomial factor, the operation sign is the same asin the expression that is being factored. In the trinomial factor, theoperation sign before the middle term is the opposite of the sign in theexpression that is being factored. The sign before the last term is alwaysa plus.

prime


Using Patterns to FactorStudy the patterns below for factoring the sum and the difference of cubes.

a3 % b3 " (a % b)(a2 ! ab % b2)a3 ! b3 " (a ! b)(a2 % ab % b2)

This pattern can be extended to other odd powers. Study these examples.

Factor a5 $ b5.Extend the first pattern to obtain a5 % b5 " (a % b)(a4 ! a3b % a2b2 ! ab3 % b4).Check: (a % b)(a4 ! a3b % a2b2 ! ab3 % b4) " a5 ! a4b % a3b2 ! a2b3 % ab4

% a4b ! a3b2 % a2b3 ! ab4 % b5

" a5 % b5

Factor a5 ! b5.Extend the second pattern to obtain a5 ! b5 " (a ! b)(a4 % a3b % a2b2 % ab3 % b4).Check: (a ! b)(a4 % a3b % a2b2 % ab3 % b4) " a5 % a4b % a3b2 % a2b3 % ab4

! a4b ! a3b2 ! a2b3 ! ab4 ! b5

" a5 ! b5

In general, if n is an odd integer, when you factor an % bn or an ! bn, one factor will beeither (a % b) or (a ! b), depending on the sign of the original expression. The other factorwill have the following properties:• The first term will be an ! 1 and the last term will be bn ! 1.• The exponents of a will decrease by 1 as you go from left to right.• The exponents of b will increase by 1 as you go from left to right.• The degree of each term will be n ! 1.• If the original expression was an % bn, the terms will alternately have % and ! signs.• If the original expression was an ! bn, the terms will all have % signs.

Use the patterns above to factor each expression.

1. a7 % b7

2. c9 ! d9

3. e11 % f 11

To factor x10 ! y10, change it to (x5 $ y5)(x5 ! y5) and factor each binomial. Usethis approach to factor each expression.

4. x10 ! y10

5. a14 ! b14

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-45-4

Example 1Example 1

Example 2Example 2

Study Guide and InterventionRoots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

on

5-5

Simplify Radicals

Square Root For any real numbers a and b, if a2 " b, then a is a square root of b.

nth Root For any real numbers a and b, and any positive integer n, if an " b, then a is an n throot of b.

1. If n is even and b , 0, then b has one positive root and one negative root.Real nth Roots of b, 2. If n is odd and b , 0, then b has one positive root.n!b", !

n!b" 3. If n is even and b ) 0, then b has no real roots.4. If n is odd and b ) 0, then b has one negative root.

Simplify !49z8".

#49z8$ " #(7z4)2$ " 7z4

z4 must be positive, so there is no need totake the absolute value.

Simplify !3!(2a !" 1)6"!

3#(2a !$ 1)6$ " !3#[(2a !$ 1)2]3$ " (2a ! 1)2

Example 1Example 1 Example 2Example 2

ExercisesExercises

Simplify.

1. #81$ 2. 3#!343$ 3. #144p6$9 !7 12|p3|

4. -#4a10$ 5. 5#243p1$0$ 6. !3#m6n9$

(2a5 3p2 !m2n3

7. 3#!b12$ 8. #16a10$b8$ 9. #121x6$!b4 4|a5|b4 11|x3|

10. #(4k)4$ 11. -#169r4$ 12. !3#!27p$6$

16k2 (13r 2 3p2

13. !#625y2$z4$ 14. #36q34$ 15. #100x2$y4z6$!25|y |z2 6|q17| 10|x |y2|z3|

16. 3#!0.02$7$ 17. !#!0.36$ 18. #0.64p$10$!0.3 not a real number 0.8|p5|

19. 4#(2x)8$ 20. #(11y2)$4$ 21. 3#(5a2b)$6$4x2 121y4 25a4b2

22. #(3x !$ 1)2$ 23. 3#(m !$5)6$ 24. #36x2 !$ 12x %$ 1$|3x ! 1| (m ! 5)2 |6x ! 1|


Approximate Radicals with a Calculator

Irrational Number a number that cannot be expressed as a terminating or a repeating decimal

Radicals such as #2$ and #3$ are examples of irrational numbers. Decimal approximationsfor irrational numbers are often used in applications. These approximations can be easilyfound with a calculator.

Approximate 5!18.2" with a calculator.

5#18.2$ & 1.787

Use a calculator to approximate each value to three decimal places.

1. #62$ 2. #1050$ 3. 3#0.054$7.874 32.404 0.378

4. !4#5.45$ 5. #5280$ 6. #18,60$0$

!1.528 72.664 136.382

7. #0.095$ 8. 3#!15$ 9. 5#100$0.308 !2.466 2.512

10. 6#856$ 11. #3200$ 12. #0.05$3.081 56.569 0.224

13. #12,50$0$ 14. #0.60$ 15. !4#500$

111.803 0.775 !4.729

16. 3#0.15$ 17. 6#4200$ 18. #75$0.531 4.017 8.660

19. LAW ENFORCEMENT The formula r " 2#5L$ is used by police to estimate the speed rin miles per hour of a car if the length L of the car’s skid mark is measures in feet.Estimate to the nearest tenth of a mile per hour the speed of a car that leaves a skidmark 300 feet long. 77.5 mi/h

20. SPACE TRAVEL The distance to the horizon d miles from a satellite orbiting h milesabove Earth can be approximated by d " #8000h$ % h2$. What is the distance to thehorizon if a satellite is orbiting 150 miles above Earth? about 1100 ft


Roots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

ExampleExample

ExercisesExercises

Skills PracticeRoots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

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1. #230$ 15.166 2. #38$ 6.164

3. !#152$ !12.329 4. #5.6$ 2.366

5. 3#88$ 4.448 6. 3#!222$ !6.055

7. !4#0.34$ !0.764 8. 5#500$ 3.466

Simplify.

9. -#81$ (9 10. #144$ 12

11. #(!5)2$ 5 12. #!52$ not a real number

13. #0.36$ 0.6 14. !'( !

15. 3#!8$ !2 16. !3#27$ !3

17. 3#0.064$ 0.4 18. 5#32$ 2

19. 4#81$ 3 20. #y2$ |y |

21. 3#125s3$ 5s 22. #64x6$ 8|x3|

23. 3#!27a$6$ !3a2 24. #m8n4$ m4n2

25. !#100p4$q2$ !10p2|q | 26. 4#16w4v$8$ 2|w |v2

27. #(!3c)4$ 9c2 28. #(a % b$)2$ |a $ b |

2"3

4&9



1. #7.8$ 2. !#89$ 3. 3#25$ 4. 3#!4$2.793 !9.434 2.924 !1.587

5. 4#1.1$ 6. 5#!0.1$ 7. 6#5555$ 8. 4#(0.94)$2$1.024 !0.631 4.208 0.970

Simplify.

9. #0.81$ 10. !#324$ 11. !4#256$ 12. 6#64$

0.9 !18 !4 213. 3#!64$ 14. 3#0.512$ 15. 5#!243$ 16. !

4#1296$!4 0.8 !3 !6

17. 5'( 18. 5#243x1$0$ 19. #(14a)2$ 20. #!(14a$)2$ not a

!"43" 3x2 14|a| real number

21. #49m2t$8$ 22. '( 23. 3#!64r6$w15$ 24. #(2x)8$

7|m |t4 !4r2w5 16x4

25. !4#625s8$ 26. 3#216p3$q9$ 27. #676x4$y6$ 28. 3#!27x9$y12$

!5s2 6pq3 26x2|y3| !3x3y4

29. !#144m$8n6$ 30. 5#!32x5$y10$ 31. 6#(m %$4)6$ 32. 3#(2x %$ 1)3$!12m4|n3| !2xy2 |m $ 4| 2x $ 1

33. !#49a10$b16$ 34. 4#(x ! 5$)8$ 35. 3#343d6$ 36. #x2 % 1$0x % 2$5$!7|a5|b8 (x ! 5)2 7d2 |x $ 5|

37. RADIANT TEMPERATURE Thermal sensors measure an object’s radiant temperature,which is the amount of energy radiated by the object. The internal temperature of an object is called its kinetic temperature. The formula Tr " Tk

4#e$ relates an object’s radianttemperature Tr to its kinetic temperature Tk. The variable e in the formula is a measureof how well the object radiates energy. If an object’s kinetic temperature is 30°C and e " 0.94, what is the object’s radiant temperature to the nearest tenth of a degree?29.5)C

38. HERO’S FORMULA Salvatore is buying fertilizer for his triangular garden. He knowsthe lengths of all three sides, so he is using Hero’s formula to find the area. Hero’sformula states that the area of a triangle is #s(s !$a)(s !$ b)(s !$ c)$, where a, b, and c arethe lengths of the sides of the triangle and s is half the perimeter of the triangle. If thelengths of the sides of Salvatore’s garden are 15 feet, 17 feet, and 20 feet, what is thearea of the garden? Round your answer to the nearest whole number. 124 ft2

4|m|"5

16m2&25

!1024&243

Practice (Average)

Roots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5

Reading to Learn MathematicsRoots of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-55-5


Less

on

5-5

Pre-Activity How do square roots apply to oceanography?


Suppose the length of a wave is 5 feet. Explain how you would estimate thespeed of the wave to the nearest tenth of a knot using a calculator. (Do notactually calculate the speed.) Sample answer: Using a calculator,find the positive square root of 5. Multiply this number by 1.34.Then round the answer to the nearest tenth.

Reading the Lesson

1. For each radical below, identify the radicand and the index.

a. 3#23$ radicand: index:

b. #15x2$ radicand: index:

c. 5#!343$ radicand: index:

2. Complete the following table. (Do not actually find any of the indicated roots.)

Number Number of Positive Number of Negative Number of Positive Number of NegativeSquare Roots Square Roots Cube Roots Cube Roots

27 1 1 1 0

!16 0 0 0 1

3. State whether each of the following is true or false.

a. A negative number has no real fourth roots. true

b. -#121$ represents both square roots of 121. true

c. When you take the fifth root of x5, you must take the absolute value of x to identifythe principal fifth root. false


4. What is an easy way to remember that a negative number has no real square roots buthas one real cube root? Sample answer: The square of a positive or negativenumber is positive, so there is no real number whose square is negative.However, the cube of a negative number is negative, so a negativenumber has one real cube root, which is a negative number.

5!343

215x2

323


Approximating Square RootsConsider the following expansion.

(a % &2ba&)2

" a2 % &22aab

& % &4ba

2

2&

" a2 % b % &4ba

2

2&

Think what happens if a is very great in comparison to b. The term &4ba

2

2& is very small and can be disregarded in an approximation.

(a % &2ba&)2

) a2 % b

a % &2ba& ) #a2 % b$

Suppose a number can be expressed as a2 % b, a , b. Then an approximate value

of the square root is a % &2ba&. You should also see that a ! &2

ba& ) #a2 ! b$.

Use the formula !a2 ("b" # a ( "2ba" to approximate !101" and !622".

a. #101$ " #100 %$ 1$ " #102 %$ 1$ b. #622$ " #625 !$ 3$ " #252 !$ 3$

Let a " 10 and b " 1. Let a " 25 and b " 3.

#101$ ) 10 % &2(110)& #622$ ) 25 ! &2(

325)&

) 10.05 ) 24.94

Use the formula to find an approximation for each square root to the nearest hundredth. Check your work with a calculator.

1. #626$ 2. #99$ 3. #402$

4. #1604$ 5. #223$ 6. #80$

7. #4890$ 8. #2505$ 9. #3575$

10. #1,441$,100$ 11. #290$ 12. #260$

13. Show that a ! &2ba& ) #a2 ! b$ for a , b. $

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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ExampleExample

Study Guide and InterventionRadical Expressions

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Simplify Radical Expressions

For any real numbers a and b, and any integer n , 1:Product Property of Radicals 1. if n is even and a and b are both nonnegative, then n#ab$ "

n#a$ $n#b$.

2. if n is odd, then n#ab$ "n#a$ $

n#b$.

To simplify a square root, follow these steps:1. Factor the radicand into as many squares as possible.2. Use the Product Property to isolate the perfect squares.3. Simplify each radical.

Quotient Property of RadicalsFor any real numbers a and b # 0, and any integer n , 1, n'( " , if all roots are defined.

To eliminate radicals from a denominator or fractions from a radicand, multiply thenumerator and denominator by a quantity so that the radicand has an exact root.

n#a$&n#b$

a&b

Simplify 3!!16a"5b7".

3#!16a$5b7$ "3#(!2)3$$ 2 $ a$3 $ a2$$ (b2)3$ $ b$

" !2ab2 3#2a2b$

Simplify %&.

'( " Quotient Property

" Factor into squares.

" Product Property

" Simplify.

" $ Rationalize thedenominator.

" Simplify.2| x|#10xy$&&

15y3

#5y$&#5y$

2| x|#2x$&3y2#5y$

2| x|#2x$&3y2#5y$

#(2x)2$ $ #2x$&&#(3y2)2$ $ #5y$

#(2x)2 $$ 2x$&&#(3y2)2$ $ 5y$

#8x3$&#45y5$

8x3&45y5

8x3"45y5Example 1Example 1 Example 2Example 2

ExercisesExercises

Simplify.

1. 5#54$ 15!6" 2. 4#32a9b$20$ 2a2|b5| 4!2a" 3. #75x4y$7$ 5x2y3!5y"

4. '( 5. '( 6. 3'( pq 3!5p2"""10

p5q3&40

|a3|b!2b"""14a6b3&98

6!5""2536&125


Operations with Radicals When you add expressions containing radicals, you canadd only like terms or like radical expressions. Two radical expressions are called likeradical expressions if both the indices and the radicands are alike.

To multiply radicals, use the Product and Quotient Properties. For products of the form (a#b$ % c#d$) $ (e#f$ % g#h$), use the FOIL method. To rationalize denominators, use conjugates. Numbers of the form a#b$ % c#d$ and a#b$ ! c#d$, where a, b, c, and d arerational numbers, are called conjugates. The product of conjugates is always a rationalnumber.

Simplify 2!50" $ 4!500" ! 6!125".

2#50$ % 4#500$ ! 6#125$ " 2#52 $ 2$ % 4#102 $ 5$ ! 6#52 $ 5$ Factor using squares.

" 2 $ 5 $ #2$ % 4 $ 10 $ #5$ ! 6 $ 5 $ #5$ Simplify square roots.

" 10#2$ % 40#5$ ! 30#5$ Multiply.

" 10#2$ % 10#5$ Combine like radicals.


Radical Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6

Example 1Example 1

Simplify (2!3" ! 4!2")(!3" $ 2!2").(2#3$ ! 4#2$)(#3$ % 2#2$)

" 2#3$ $ #3$ % 2#3$ $ 2#2$ ! 4#2$ $ #3$ ! 4#2$ $ 2#2$" 6 % 4#6$ ! 4#6$ ! 16" !10

Simplify .

" $

"

"

"11 ! 5#5$&&4

6 ! 5#5$ % 5&&9 ! 5

6 ! 2#5$ ! 3#5$ % (#5$ )2&&&

32 ! (#5$)2

3 ! #5$&3 ! #5$

2 ! #5$&3 % #5$

2 ! #5$&3 % #5$

2 ! !5""3 $ !5"


ExercisesExercises

Simplify.

1. 3#2$ % #50$ ! 4#8$ 2. #20$ % #125$ ! #45$ 3. #300$ ! #27$ ! #75$

0 4!5" 2!3"

4. 3#81$ $3#24$ 5. 3#2$( 3#4$ %

3#12$) 6. 2#3$(#15$ % #60$ )

6 3!9" 2 $ 2 3!3" 18!5"

7. (2 % 3#7$)(4 % #7$) 8. (6#3$ ! 4#2$)(3#3$ % #2$) 9. (4#2$ ! 3#5$)(2#20 %$5$)

29 $ 14!7" 46 ! 6!6" 40!2" ! 30!5"

10. 5 11. 5 $ 3!2" 12. 13!3" ! 23""115 ! 3#3$&&1 % 2#3$

4 % #2$&2 ! #2$

5#48$ % #75$&&

5#3$

Skills PracticeRadical Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6


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Simplify.

1. #24$ 2!6" 2. #75$ 5!3"

3. 3#16$ 2 3!2" 4. !4#48$ !2 4!3"

5. 4#50x5$ 20x2!2x" 6. 4#64a4b$4$ 2|ab | 4!4"

7. 3'(! d2f5 ! f 3!d 2f 2" 8. '(s2t |s |!t"

9. !'( ! 10. 3'(

11. '(&25gz3& 12. (3#3$)(5#3$) 45

13. (4#12$ )(3#20$ ) 48!15" 14. #2$ % #8$ % #50$ 8!2"

15. #12$ ! 2#3$ % #108$ 6!3" 16. 8#5$ ! #45$ ! #80$ !5"

17. 2#48$ ! #75$ ! #12$ !3" 18. (2 % #3$)(6 ! #2$) 12 ! 2!2" $ 6!3" ! !6"

19. (1 ! #5$)(1 % #5$) !4 20. (3 ! #7$)(5 % #2$) 15 $ 3!2" ! 5!7" ! !14"

21. (#2$ ! #6$)2 8 ! 4!3" 22.

23. 24. 40 $ 5!6"""585

&8 ! #6$

12 ! 4!2"""74

&3 % #2$

21 $ 3!2"""473

&7 ! #2$

g!10gz"""5z

3!6""32&9

!21""73&7

5"6

25&36

1"2

1&8


Simplify.

1. #540$ 6!15" 2. 3#!432$ !6 3!2" 3. 3#128$ 4 3!2"

4. !4#405$ !3 4!5" 5. 3#!500$0$ !10 3!5" 6. 5#!121$5$ !3 5!5"

7. 3#125t6w$2$ 5t 2 3!w2" 8. 4#48v8z$13$ 2v2z3 4!3z" 9. 3#8g3k8$ 2gk2 3!k2"

10. #45x3y$8$ 3xy4!5x" 11. '( 12. 3'( 3!9"

13. '(c4d7 c2d 3!2d" 14. '( 15. 4'(16. (3#15$ )(!4#45$ ) 17. (2#24$ )(7#18$ ) 18. #810$ % #240$ ! #250$

!180!3" 168!3" 4!10" $ 4!15"

19. 6#20$ % 8#5$ ! 5#45$ 20. 8#48$ ! 6#75$ % 7#80$ 21. (3#2$ % 2#3$)2

5!5" 2!3" $ 28!5" 30 $ 12!6"

22. (3 ! #7$)2 23. (#5$ ! #6$)(#5$ % #2$) 24. (#2$ % #10$ )(#2$ ! #10$ )16 ! 6!7" 5 $ !10" ! !30" ! 2!3" !8

25. (1 % #6$)(5 ! #7$) 26. (#3$ % 4#7$)2 27. (#108$ ! 6#3$)2

5 ! !7" $ 5!6" ! !42" 115 $ 8!21" 0

28. !15" $ 2!3" 29. 6!2" $ 6 30.

31. 32. 27 $ 11!6" 33.

34. BRAKING The formula s " 2#5!$ estimates the speed s in miles per hour of a car whenit leaves skid marks ! feet long. Use the formula to write a simplified expression for s if ! " 85. Then evaluate s to the nearest mile per hour. 10!17"; 41 mi/h

35. PYTHAGOREAN THEOREM The measures of the legs of a right triangle can berepresented by the expressions 6x2y and 9x2y. Use the Pythagorean Theorem to find asimplified expression for the measure of the hypotenuse. 3x2 |y |!13"

6 $ 5!x" $ x""4 ! x3 % #x$&2 ! #x$

3 % #6$&&5 ! #24$

8 $ 5!2"""2

3 % #2$&2 ! #2$

17 ! !3"""135 % #3$&4 % #3$

6&#2$ ! 1

#3$&#5$ ! 2

4!72a""3a8

&9a33a2!a""8b2

9a5&64b4

1"16

1&128

216&24

!11""311&9

Practice (Average)

Radical Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6

Reading to Learn MathematicsRadical Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6


Less

on

5-6

Pre-Activity How do radical expressions apply to falling objects?


Describe how you could use the formula given in your textbook and acalculator to find the time, to the nearest tenth of a second, that it wouldtake for the water balloons to drop 22 feet. (Do not actually calculate thetime.) Sample answer: Multiply 22 by 2 (giving 44) and divideby 32. Use the calculator to find the square root of the result.Round this square root to the nearest tenth.

Reading the Lesson

1. Complete the conditions that must be met for a radical expression to be in simplified form.

• The n is as as possible.

• The contains no (other than 1) that are nth

powers of a(n) or polynomial.

• The radicand contains no .

• No appear in the .

2. a. What are conjugates of radical expressions used for? to rationalize binomialdenominators

b. How would you use a conjugate to simplify the radical expression ?

Multiply numerator and denominator by 3 $ !2".c. In order to simplify the radical expression in part b, two multiplications are

necessary. The multiplication in the numerator would be done by themethod, and the multiplication in the denominator would be done by finding the

of .


3. One way to remember something is to explain it to another person. When rationalizing the

denominator in the expression , many students think they should multiply numerator

and denominator by . How would you explain to a classmate why this is incorrect

and what he should do instead. Sample answer: Because you are working withcube roots, not square roots, you need to make the radicand in thedenominator a perfect cube, not a perfect square. Multiply numerator and denominator by to make the denominator 3!8", which equals 2.

3!4""3!4"

3#2$&3#2$

1&3#2$

squarestwodifference

FOIL

1 % #2$&3 ! #2$

denominatorradicalsfractions

integerfactorsradicand

smallindex


Special Products with RadicalsNotice that (#3$)(#3$) " 3, or (#3$)2 " 3.

In general, (#x$)2 " x when x . 0.

Also, notice that (#9$)(#4$) " #36$.

In general, (#x$)(#y$) " #xy$ when x and y are not negative.

You can use these ideas to find the special products below.

(#a$ % #b$)(#a$ ! #b$) " (#a$)2 ! (#b$)2 " a ! b

(#a$ % #b$)2 " (#a$)2 % 2#ab$ % (#b$)2 " a % 2#ab$ % b

(#a$ ! #b$)2 " (#a$)2 ! 2#ab$ % (#b$)2 " a ! 2#ab$ % b

Find the product: (!2" $ !5")(!2" ! !5").(#2$ % #5$)(#2$ ! #5$) " (#2$)2 ! (#5$)2 " 2 ! 5 " !3

Evaluate (!2" $ !8")2.

(#2$ % #8$)2 " (#2$)2 % 2#2$#8$ % (#8$)2

" 2 % 2#16$ % 8 " 2 % 2(4) % 8 " 2 % 8 % 8 " 18

Multiply.

1. (#3$ ! #7$)(#3$ % #7$) 2. (#10$ % #2$)(#10$ ! #2$)

3. (#2x$ ! #6$)(#2x$ ! #6$) 4. (#3$ ! 27)2

5. (#1000$ % #10$ )2 6. (#y$ % #5$)(#y$ ! #5$)

7. (#50$ ! #x$)2 8. (#x$ % 20)2

You can extend these ideas to patterns for sums and differences of cubes.Study the pattern below.

( 3#8$ ! 3#x$)( 3#82$ %

3#8x$ %3#x2$) "

3#83$ ! 3#x3$ " 8 ! x

Multiply.

9. ( 3#2$ !3#5$)( 3#22$ %

3#10$ %3#52$)

10. ( 3#y$ %3#w$)( 3#y2$ !

3#yw$ %3#w2$ )

11. ( 3#7$ %3#20$ )( 3#72$ !

3#140$ %3#202$ )

12. ( 3#11$ !3#8$)( 3#112$ %

3#88$ %3#82$)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-65-6

Example 1Example 1

Example 2Example 2

Study Guide and InterventionRational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7


Less

on

5-7

Rational Exponents and Radicals

Definition of b"n1" For any real number b and any positive integer n,

b&1n&

"n#b$, except when b ) 0 and n is even.

Definition of b"mn" For any nonzero real number b, and any integers m and n, with n , 1,

b&mn&

"n#bm$ " ( n#b$)m, except when b ) 0 and n is even.

Write 28"12" in radical form.

Notice that 28 , 0.

28&12

&" #28$" #22 $ 7$" #22$ $ #7$" 2#7$

Evaluate $ '"13"

.

Notice that !8 ) 0, !125 ) 0, and 3 is odd.

! "&13

&"

"

"2&5

!2&!5

3#!8$&3#!125$

!8&!125

!8"!125


ExercisesExercises

Write each expression in radical form.

1. 11&17

&2. 15

&13

&3. 300

&32

&

7!11" 3!15" !3003"

Write each radical using rational exponents.

4. #47$ 5. 3#3a5b2$ 6. 4#162p5$

47"12" 3"

13"a"

53"b"

23" 3 * 2"

14"

* p"54"

Evaluate each expression.

7. !27&23

&8. 9. (0.0004)

&12

&

9 0.02

10. 8&23

&$ 4

&32

&11. 12.

32 1"2

1"4

16!&12

&

&(0.25)

&12

&

144!&12

&

&

27!&13

&

1"10

5!&12

&

&2#5$


Simplify Expressions All the properties of powers from Lesson 5-1 apply to rationalexponents. When you simplify expressions with rational exponents, leave the exponent inrational form, and write the expression with all positive exponents. Any exponents in thedenominator must be positive integers

When you simplify radical expressions, you may use rational exponents to simplify, but youranswer should be in radical form. Use the smallest index possible.


Rational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7

Simplify y"23"

* y"38".

y&23

&$ y

&38

&" y

&23

& % &38

&" y

&2254&

Simplify 4!144x6".

4#144x6$ " (144x6)&14

&

" (24 $ 32 $ x6)&14

&

" (24)&14

&$ (32)&

14

&$ (x6)&

14

&

" 2 $ 3&12

&$ x

&32

&" 2x $ (3x)&

12

&" 2x#3x$


ExercisesExercises

Simplify each expression.

1. x&45

&$ x

&65

& 2. !y&23

&" &34

& 3. p&45

&$ p

&170&

x2 y "12" p"

32"

4. !m!&65

&" &25

& 5. x!&38

&$ x

&43

& 6. !s!&16

&"!&43

&

x"2234" s"

29"

7. 8. !a&23

&" &65

&$ !a&

25

&"3 9.

p"23" a2

10. 6#128$ 11. 4#49$ 12. 5#288$

2!2" !7" 2 5!9"

13. #32$ $ 3#16$ 14. 3#25$ $ #125$ 15. 6#16$

48!2" 25 6!5" 3!4"

16. 17. #$3#48$ 18.

6!48" !a" 6!b5""bx!3" !

6!35"""6

a3#b4$

&#ab3$

x !3#3$

&#12$

x"56

"

"x

x!&12

&

&x!&

13

&

p&p

&13

&

1"m3

Skills PracticeRational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7


Less

on

5-7


1. 3&16

& 6!3" 2. 8&15

& 5!8"

3. 12&23

& 3!122" or ( 3!12")24. (s3)&

35

& s 5!s4"


5. #51$ 51"12"

6. 3#37$ 37"13"

7. 4#153$ 15"34"

8. 3#6xy2$ 6"13"x"

13"y"

23"


9. 32&15

& 2 10. 81&14

& 3

11. 27!&13

& 12. 4!&12

&

13. 16&32

& 64 14. (!243)&45

& 81

15. 27&13

&$ 27&

53

& 729 16. ! "&32

&


17. c&152&

$ c&35

& c3 18. m&29

&$ m

&196& m2

19. !q&12

&"3 q"32"

20. p!&15

&

21. x!&161& 22. x"1

52"

23. 24.

25.12#64$ !2" 26. 8#49a8b$2$ |a | 4!7b"

n"23

"

"nn

&13

&

&

n&16

&$ n

&12

&

y"14

"

"yy!&

12

&

&

y&14

&

x&23

&

&

x&14

&

x"151"

"x

p"45

"

"p

8"27

4&9

1"2

1"3



1. 5&13

& 2. 6&25

& 3. m&47

& 4. (n3)&25

&

3!5" 5!62" or ( 5!6")2 7!m4" or ( 7!m")4 n 5!n"


5. #79$ 6. 4#153$ 7. 3#27m6n$4$ 8. 5#2a10b$

79"12" 153"

14" 3m2n"

43" 5 * 2"

12" |a5 |b"

12"


9. 81&14

& 3 10. 1024!&15

& 11. 8!&53

&

12. !256!&34

& ! 13. (!64)!&23

& 14. 27&13

&$ 27&

43

& 243

15. ! "&23

& 16. 17. !25&12

&"!!64!&13

&" !


18. g&47

&$ g

&37

& g 19. s&34

&$ s

&143& s4 20. !u!&

13

&"!&45

& u"145" 21. y!&

12

&

22. b!&35

& 23. q "15"

24. 25.

26.10#85$ 2!2" 27. #12$ $

5#123$ 28. 4#6$ $ 3 4#6$ 29.

1210!12" 3!6"

30. ELECTRICITY The amount of current in amperes I that an appliance uses can be

calculated using the formula I " ! "&12

&, where P is the power in watts and R is the

resistance in ohms. How much current does an appliance use if P " 500 watts and R " 10 ohms? Round your answer to the nearest tenth. 7.1 amps

31. BUSINESS A company that produces DVDs uses the formula C " 88n&13

&% 330 to

calculate the cost C in dollars of producing n DVDs per day. What is the company’s costto produce 150 DVDs per day? Round your answer to the nearest dollar. $798

P&R

a!3b""3ba

&#3b$

2z $ 2z"12

"

""z ! 12z

&12

&

&

z&12

&! 1

t"11

12"

"5t

&23

&

&

5t&12

&$ t!&

34

&

q&35

&

&q

&25

&

b"25

"

"b

y"12

"

"y

5"4

16"49

64&23

&

&

343&23

&

25"36

125&216

1"16

1"64

1"32

1"4

Practice (Average)

Rational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7

Reading to Learn MathematicsRational Exponents

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7


Less

on

5-7

Pre-Activity How do rational exponents apply to astronomy?


The formula in the introduction contains the exponent . What do you think it might mean to raise a number to the power?

Sample answer: Take the fifth root of the number and square it.

Reading the Lesson

1. Complete the following definitions of rational exponents.

• For any real number b and for any positive integer n, b&n1

&" except

when b and n is .

• For any nonzero real number b, and any integers m and n, with n ,

b&mn

&" " , except when b and

n is .

2. Complete the conditions that must be met in order for an expression with rationalexponents to be simplified.

• It has no exponents.

• It has no exponents in the .

• It is not a fraction.

• The of any remaining is the number possible.

3. Margarita and Pierre were working together on their algebra homework. One exercise

asked them to evaluate the expression 27&43

&. Margarita thought that they should raise 27 to the fourth power first and then take the cube root of the result. Pierre thought thatthey should take the cube root of 27 first and then raise the result to the fourth power.Whose method is correct? Both methods are correct.


4. Some students have trouble remembering which part of the fraction in a rationalexponent gives the power and which part gives the root. How can your knowledge ofinteger exponents help you to keep this straight? Sample answer: An integer exponent can be written as a rational exponent. For example, 23 & 2"

31".

You know that this means that 2 is raised to the third power, so thenumerator must give the power, and, therefore, the denominator mustgive the root.

leastradicalindexcomplex

denominatorfractionalnegative

even+ 0( n!b")mn

!bm", 1

even+ 0

n!b"

2&5

2&5


Lesser-Known Geometric FormulasMany geometric formulas involve radical expressions.

Make a drawing to illustrate each of the formulas given on this page.Then evaluate the formula for the given value of the variable. Round answers to the nearest hundredth.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-75-7

1. The area of an isosceles triangle. Twosides have length a; the other side haslength c. Find A when a " 6 and c " 7.

A " &4c

&#4a2 !$ c2$A ( 17.06

3. The area of a regular pentagon with aside of length a. Find A when a " 4.

A " &a42&#25 %$10#5$$

A ( 27.53

5. The volume of a regular tetrahedronwith an edge of length a. Find V when a " 2.

V " &1a23&#2$

V ( 0.94

7. Heron’s Formula for the area of a triangle uses the semi-perimeter s,where s " &

a %2b % c&. The sides of the

triangle have lengths a, b, and c. Find Awhen a " 3, b " 4, and c " 5.

A " #s(s !$a)(s !$ b)(s !$ c)$A & 6

2. The area of an equilateral triangle witha side of length a. Find A when a " 8.

A " &a42&#3$

A ( 27.71

4. The area of a regular hexagon with aside of length a. Find A when a " 9.

A " &32a2&#3$

A ( 210.44

6. The area of the curved surface of a rightcone with an altitude of h and radius ofbase r. Find S when r " 3 and h " 6.

S " +r#r2 % h$2$S ( 63.22

8. The radius of a circle inscribed in a giventriangle also uses the semi-perimeter.Find r when a " 6, b " 7, and c " 9.

r "

r ( 1.91

#s(s !$a)(s !$ b)(s !$ c)$&&&s

Study Guide and InterventionRadical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8


Less

on

5-8

Solve Radical Equations The following steps are used in solving equations that havevariables in the radicand. Some algebraic procedures may be needed before you use thesesteps.

Step 1 Isolate the radical on one side of the equation.Step 2 To eliminate the radical, raise each side of the equation to a power equal to the index of the radical.Step 3 Solve the resulting equation.Step 4 Check your solution in the original equation to make sure that you have not obtained any extraneous roots.

Solve 2!4x $"8" ! 4 & 8.

2#4x % 8$ ! 4 " 8 Original equation

2#4x % 8$ " 12 Add 4 to each side.

#4x % 8$ " 6 Isolate the radical.

4x % 8 " 36 Square each side.

4x " 28 Subtract 8 from each side.

x " 7 Divide each side by 4.

Check2#4(7) %$ 8$ ! 4 ! 8

2#36$ ! 4 ! 82(6) ! 4 ! 8

8 " 8The solution x " 7 checks.

Solve !3x $"1" & !5x" ! 1.

#3x % 1$ " #5x$ ! 1 Original equation

3x % 1 " 5x ! 2#5x$ % 1 Square each side.

2#5x$ " 2x Simplify.

#5x$ " x Isolate the radical.

5x " x2 Square each side.

x2 ! 5x " 0 Subtract 5x from each side.

x(x ! 5) " 0 Factor.

x " 0 or x " 5Check#3(0) %$ 1$ " 1, but #5(0)$ ! 1 " !1, so 0 isnot a solution.#3(5) %$ 1$ " 4, and #5(5)$ ! 1 " 4, so thesolution is x " 5.


ExercisesExercises

Solve each equation.

1. 3 % 2x#3$ " 5 2. 2#3x % 4$ % 1 " 15 3. 8 % #x % 1$ " 2

15 no solution

4. #5 ! x$ ! 4 " 6 5. 12 % #2x ! 1$ " 4 6. #12 !$x$ " 0

!95 no solution 12

7. #21$ ! #5x ! 4$ " 0 8. 10 ! #2x$ " 5 9. #x2 % 7$x$ " #7x ! 9$

5 12.5 no solution

10. 4 3#2x % 1$1$ ! 2 " 10 11. 2#x % 11$ " #x % 2$ % #3x ! 6$ 12. #9x ! 1$1$ " x % 1

8 14 3, 4

!3""3


Solve Radical Inequalities A radical inequality is an inequality that has a variablein a radicand. Use the following steps to solve radical inequalities.

Step 1 If the index of the root is even, identify the values of the variable for which the radicand is nonnegative.Step 2 Solve the inequality algebraically.Step 3 Test values to check your solution.

Solve 5 ! !20x $" 4" - !3.


Radical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8

ExampleExampleSince the radicand of a square rootmust be greater than or equal tozero, first solve20x % 4 . 0.20x % 4 . 0

20x . !4

x . !1&5

Now solve 5 ! #20x %$ 4$ . ! 3.

5 ! #20x %$ 4$ . !3 Original inequality

#20x %$ 4$ ( 8 Isolate the radical.

20x % 4 ( 64 Eliminate the radical by squaring each side.

20x ( 60 Subtract 4 from each side.

x ( 3 Divide each side by 20.

It appears that ! ( x ( 3 is the solution. Test some values.

x & !1 x & 0 x & 4

#20(!1$) % 4$ is not a real 5 ! #20(0)$% 4$ " 3, so the 5 ! #20(4)$% 4$ & !4.2, sonumber, so the inequality is inequality is satisfied. the inequality is notnot satisfied. satisfied

Therefore the solution ! ( x ( 3 checks.

Solve each inequality.

1. #c ! 2$ % 4 . 7 2. 3#2x ! 1$ % 6 ) 15 3. #10x %$ 9$ ! 2 , 5

c - 11 . x + 5 x , 4

4. 5 3#x % 2$ ! 8 ) 2 5. 8 ! #3x % 4$ . 3 6. #2x % 8$ ! 4 , 2

x + 6 ! . x . 7 x , 14

7. 9 ! #6x % 3$ . 6 8. ( 4 9. 2#5x ! 6$ ! 1 ) 5

! . x . 1 x - 8 . x + 3

10. #2x % 1$2$ % 4 . 12 11. #2d %$1$ % #d$ ( 5 12. 4#b % 3$ ! #b ! 2$ . 10

x - 26 0 . d . 4 b - 6

6"5

1"2

20&&#3x % 1$

4"3

1"2

1&5

1&5

ExercisesExercises

Skills PracticeRadical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8


Less

on

5-8

Solve each equation or inequality.

1. #x$ " 5 25 2. #x$ % 3 " 7 16

3. 5#j$ " 1 4. v&12

&% 1 " 0 no solution

5. 18 ! 3y&12

&" 25 no solution 6. 3#2w$ " 4 32

7. #b ! 5$ " 4 21 8. #3n %$1$ " 5 8

9. 3#3r ! 6$ " 3 11 10. 2 % #3p %$7$ " 6 3

11. #k ! 4$ ! 1 " 5 40 12. (2d % 3)&13

&" 2

13. (t ! 3)&13

&" 2 11 14. 4 ! (1 ! 7u)&

13

&" 0 !9

15. #3z ! 2$ " #z ! 4$ no solution 16. #g % 1$ " #2g !$7$ 8

17. #x ! 1$ " 4#x % 1$ no solution 18. 5 % #s ! 3$ ( 6 3 . s . 4

19. !2 % #3x % 3$ ) 7 !1 + x + 26 20. !#2a %$4$ . !6 !2 . a . 16

21. 2#4r ! 3$ , 10 r , 7 22. 4 ! #3x % 1$ , 3 ! + x + 0

23. #y % 4$ ! 3 . 3 y - 32 24. !3#11r %$ 3$ . !15 ! . r . 23"11

1"3

5"2

1"25


Solve each equation or inequality.

1. #x$ " 8 64 2. 4 ! #x$ " 3 1

3. #2p$ % 3 " 10 4. 4#3h$ ! 2 " 0

5. c&12

&% 6 " 9 9 6. 18 % 7h

&12

&" 12 no solution

7. 3#d % 2$ " 7 341 8. 5#w ! 7$ " 1 8

9. 6 % 3#q ! 4$ " 9 31 10. 4#y ! 9$ % 4 " 0 no solution

11. #2m !$ 6$ ! 16 " 0 131 12. 3#4m %$ 1$ ! 2 " 2

13. #8n !$5$ ! 1 " 2 14. #1 ! 4$t$ ! 8 " !6 !

15. #2t ! 5$ ! 3 " 3 16. (7v ! 2)&14

&% 12 " 7 no solution

17. (3g % 1)&12

&! 6 " 4 33 18. (6u ! 5)&

13

&% 2 " !3 !20

19. #2d !$5$ " #d ! 1$ 4 20. #4r ! 6$ " #r$ 2

21. #6x ! 4$ " #2x % 1$0$ 22. #2x % 5$ " #2x % 1$ no solution

23. 3#a$ . 12 a - 16 24. #z % 5$ % 4 ( 13 !5 . z . 76

25. 8 % #2q$ ( 5 no solution 26. #2a !$3$ ) 5 + a + 14

27. 9 ! #c % 4$ ( 6 c - 5 28. 3#x ! 1$ ) !2 x + !7

29. STATISTICS Statisticians use the formula ! " #v$ to calculate a standard deviation !,where v is the variance of a data set. Find the variance when the standard deviation is 15. 225

30. GRAVITATION Helena drops a ball from 25 feet above a lake. The formula

t " #25 !$h$ describes the time t in seconds that the ball is h feet above the water.

How many feet above the water will the ball be after 1 second? 9 ft

1&4

3"2

7"2

41"2

3"4

7"4

63"4

1"12

49"2

Practice (Average)

Radical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8

Reading to Learn MathematicsRadical Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8


Less

on

5-8

Pre-Activity How do radical equations apply to manufacturing?


Explain how you would use the formula in your textbook to find the cost ofproducing 125,000 computer chips. (Describe the steps of the calculation in theorder in which you would perform them, but do not actually do the calculation.)

Sample answer: Raise 125,000 to the power by taking the cube root of 125,000 and squaring the result (or raise 125,000 to the power by entering 125,000 ^ (2/3) on a calculator).Multiply the number you get by 10 and then add 1500.

Reading the Lesson

1. a. What is an extraneous solution of a radical equation? Sample answer: a numberthat satisfies an equation obtained by raising both sides of the originalequation to a higher power but does not satisfy the original equation

b. Describe two ways you can check the proposed solutions of a radical equation in orderto determine whether any of them are extraneous solutions. Sample answer: Oneway is to check each proposed solution by substituting it into theoriginal equation. Another way is to use a graphing calculator to graphboth sides of the original equation. See where the graphs intersect.This can help you identify solutions that may be extraneous.

2. Complete the steps that should be followed in order to solve a radical inequality.

Step 1 If the of the root is , identify the values of

the variable for which the is .

Step 2 Solve the algebraically.

Step 3 Test to check your solution.


3. One way to remember something is to explain it to another person. Suppose that yourfriend Leora thinks that she does not need to check her solutions to radical equations bysubstitution because she knows she is very careful and seldom makes mistakes in herwork. How can you explain to her that she should nevertheless check every proposedsolution in the original equation? Sample answer: Squaring both sides of anequation can produce an equation that is not equivalent to the originalone. For example, the only solution of x & 5 is 5, but the squaredequation x2 & 25 has two solutions, 5 and !5.

valuesinequality

nonnegativeradicandevenindex

2"3

2"3


Truth TablesIn mathematics, the basic operations are addition, subtraction, multiplication,division, finding a root, and raising to a power. In logic, the basic operations are the following: not (*), and (+), or (,), and implies (→).

If P and Q are statements, then *P means not P; *Q means not Q; P + Qmeans P and Q; P , Q means P or Q; and P → Q means P implies Q. The operations are defined by truth tables. On the left below is the truth table for the statement *P. Notice that there are two possible conditions for P, true (T) or false (F). If P is true, *P is false; if P is false, *P is true. Also shown are the truth tables for P + Q, P , Q, and P → Q.

P *P P Q P + Q P Q P , Q P Q P → QT F T T T T T T T T TF T T F F T F T T F F

F T F F T T F T TF F F F F F F F T

You can use this information to find out under what conditions a complex statement is true.

Under what conditions is )P * Q true?

Create the truth table for the statement. Use the information from the truth table above for P * Q to complete the last column.

P Q *P *P , QT T F TT F F FF T T TF F T T

The truth table indicates that *P , Q is true in all cases except where P is true and Q is false.

Use truth tables to determine the conditions under which each statement is true.

1. *P , *Q 2. *P → (P → Q)

3. (P , Q) , (*P + *Q) 4. (P → Q) , (Q → P)

5. (P → Q) + (Q → P) 6. (*P + *Q) → *(P , Q)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-85-8

ExampleExample

To solve a quadratic equation that does not have real solutions, you can use the fact thati2 " !1 to find complex solutions.

Solve 2x2 $ 24 & 0.

2x2 % 24 " 0 Original equation

2x2 " !24 Subtract 24 from each side.

x2 " !12 Divide each side by 2.

x " -#!12$ Take the square root of each side.

x " -2i#3$ #!12$ " #4$ $ #!1$ $ #3$

Simplify.

1. (!4 % 2i) % (6 ! 3i) 2. (5 ! i) ! (3 ! 2i) 3. (6 ! 3i) % (4 ! 2i)2 ! i 2 $ i 10 ! 5i

4. (!11 % 4i) ! (1 ! 5i) 5. (8 % 4i) % (8 ! 4i) 6. (5 % 2i) ! (!6 ! 3i)!12 $ 9i 16 11 $ 5i

7. (12 ! 5i) ! (4 % 3i) 8. (9 % 2i) % (!2 % 5i) 9. (15 ! 12i) % (11 ! 13i)8 ! 8i 7 $ 7i 26 ! 25i

10. i4 11. i6 12. i15

1 !1 !i


13. 5x2 % 45 " 0 14. 4x2 % 24 " 0 15. !9x2 " 9(3i (i!6" (i

Study Guide and InterventionComplex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9


Less

on

5-9

Add and Subtract Complex Numbers

A complex number is any number that can be written in the form a % bi, Complex Number where a and b are real numbers and i is the imaginary unit (i 2 " !1).

a is called the real part, and b is called the imaginary part.

Addition and Combine like terms.Subtraction of (a % bi) % (c % di) " (a % c) % (b % d )iComplex Numbers (a % bi) ! (c % di) " (a ! c) % (b ! d )i

Simplify (6 $ i) $ (4 ! 5i).

(6 % i) % (4 ! 5i)" (6 % 4) % (1 ! 5)i" 10 ! 4i

Simplify (8 $ 3i) ! (6 ! 2i).

(8 % 3i) ! (6 ! 2i)" (8 ! 6) % [3 ! (!2)]i" 2 % 5i


Example 3Example 3

ExercisesExercises


Multiply and Divide Complex Numbers

Multiplication of Complex Numbers Use the definition of i2 and the FOIL method:(a % bi)(c % di) " (ac ! bd ) % (ad % bc)i

To divide by a complex number, first multiply the dividend and divisor by the complexconjugate of the divisor.

Complex Conjugate a % bi and a ! bi are complex conjugates. The product of complex conjugates is always a real number.

Simplify (2 ! 5i) * (!4 $ 2i).

(2 ! 5i) $ (!4 % 2i)" 2(!4) % 2(2i) % (!5i)(!4) % (!5i)(2i) FOIL

" !8 % 4i % 20i ! 10i2 Multiply.

" !8 % 24i ! 10(!1) Simplify.

" 2 % 24i Standard form

Simplify .

" $ Use the complex conjugate of the divisor.

" Multiply.

" i2 " !1

" ! i Standard form

Simplify.

1. (2 % i)(3 ! i) 7 $ i 2. (5 ! 2i)(4 ! i) 18 ! 13i 3. (4 ! 2i)(1 ! 2i) !10i

4. (4 ! 6i)(2 % 3i) 26 5. (2 % i)(5 ! i) 11 $ 3i 6. (5 ! 3i)(!1 ! i) !8 ! 2i

7. (1 ! i)(2 % 2i)(3 ! 3i) 8. (4 ! i)(3 ! 2i)(2 % i) 9. (5 ! 2i)(1 ! i)(3 % i)12 ! 12i 31 ! 12i 16 ! 18i

10. ! i 11. ! ! i 12. ! ! 2i

13. 1 ! i 14. ! ! 2i 15. ! $ i

16. $ 17. !1 ! 2i!2" 18. $ 2i!6""3!3""3

#6$ % i#3$&&

#2$ ! i4 ! i#2$&&

i#2$3i!5""7

2"7

3 % i#5$&&3 ! i#5$

31"41

8"41

3 % 4i&4 ! 5i

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4 ! 2i&3 % i

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6 ! 9i ! 2i % 3i2&&&

4 ! 9i2

2 ! 3i&2 ! 3i

3 ! i&2 % 3i

3 ! i&2 % 3i

3 ! i"2 % 3i


Complex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeComplex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9


Less

on

5-9

Simplify.

1. #!36$ 6i 2. #!196$ 14i

3. #!81x6$ 9|x3|i 4. #!23$ $ #!46$ !23!2"

5. (3i)(!2i)(5i) 30i 6. i11 !i

7. i65 i 8. (7 ! 8i) % (!12 ! 4i) !5 ! 12i

9. (!3 % 5i) % (18 ! 7i) 15 ! 2i 10. (10 ! 4i) ! (7 % 3i) 3 ! 7i

11. (2 % i)(2 % 3i) 1 $ 8i 12. (2 % i)(3 ! 5i) 11 ! 7i

13. (7 ! 6i)(2 ! 3i) !4 ! 33i 14. (3 % 4i)(3 ! 4i) 25

15. 16.


17. 3x2 % 3 " 0 (i 18. 5x2 % 125 " 0 (5i

19. 4x2 % 20 " 0 (i!5" 20. !x2 ! 16 " 0 (4i

21. x2 % 18 " 0 (3i!2" 22. 8x2 % 96 " 0 (2i!3"

Find the values of m and n that make each equation true.

23. 20 ! 12i " 5m % 4ni 4, !3 24. m ! 16i " 3 ! 2ni 3, 8

25. (4 % m) % 2ni " 9 % 14i 5, 7 26. (3 ! n) % (7m ! 14)i " 1 % 7i 3, 2

3 $ 6i"10

3i&4 % 2i

!6 ! 8i"3

8 ! 6i&3i


Simplify.

1. #!49$ 7i 2. 6#!12$ 12i!3" 3. #!121$s8$ 11s4i

4. #!36a$3b4$ 5. #!8$ $ #!32$ 6. #!15$ $ #!25$6|a|b2i!a" !16 !5!15"

7. (!3i)(4i)(!5i) 8. (7i)2(6i) 9. i42

!60i !294i !1

10. i55 11. i89 12. (5 ! 2i) % (!13 ! 8i)!i i !8 ! 10i

13. (7 ! 6i) % (9 % 11i) 14. (!12 % 48i) % (15 % 21i) 15. (10 % 15i) ! (48 ! 30i)16 $ 5i 3 $ 69i !38 $ 45i

16. (28 ! 4i) ! (10 ! 30i) 17. (6 ! 4i)(6 % 4i) 18. (8 ! 11i)(8 ! 11i)18 $ 26i 52 !57 ! 176i

19. (4 % 3i)(2 ! 5i) 20. (7 % 2i)(9 ! 6i) 21.

23 ! 14i 75 ! 24i

22. 23. 24. !1 ! i


25. 5n2 % 35 " 0 (i!7" 26. 2m2 % 10 " 0 (i!5"

27. 4m2 % 76 " 0 (i!19" 28. !2m2 ! 6 " 0 (i!3"

29. !5m2 ! 65 " 0 (i!13" 30. x2 % 12 " 0 (4i

Find the values of m and n that make each equation true.

31. 15 ! 28i " 3m % 4ni 5, !7 32. (6 ! m) % 3ni " !12 % 27i 18, 9

33. (3m % 4) % (3 ! n)i " 16 ! 3i 4, 6 34. (7 % n) % (4m ! 10)i " 3 ! 6i 1, !4

35. ELECTRICITY The impedance in one part of a series circuit is 1 % 3j ohms and theimpedance in another part of the circuit is 7 ! 5j ohms. Add these complex numbers tofind the total impedance in the circuit. 8 ! 2 j ohms

36. ELECTRICITY Using the formula E " IZ, find the voltage E in a circuit when thecurrent I is 3 ! j amps and the impedance Z is 3 % 2j ohms. 11 $ 3j volts

3&4

2 ! 4i&1 % 3i

7 $ i"5

3 ! i&2 ! i

14 $ 16i""113

2&7 ! 8i

!5 $ 6i"2

6 % 5i&

!2i

Practice (Average)

Complex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9

Reading to Learn MathematicsComplex Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9


Less

on

5-9

Pre-Activity How do complex numbers apply to polynomial equations?


Suppose the number i is defined such that i2 " !1. Complete each equation.

2i2 " (2i)2 " i4 "

Reading the Lesson

1. Complete each statement.

a. The form a % bi is called the of a complex number.

b. In the complex number 4 % 5i, the real part is and the imaginary part is .

This is an example of a complex number that is also a(n) number.

c. In the complex number 3, the real part is and the imaginary part is .

This is example of complex number that is also a(n) number.

d. In the complex number 7i, the real part is and the imaginary part is .

This is an example of a complex number that is also a(n) number.

2. Give the complex conjugate of each number.

a. 3 % 7i

b. 2 ! i

3. Why are complex conjugates used in dividing complex numbers? The product ofcomplex conjugates is always a real number.

4. Explain how you would use complex conjugates to find (3 % 7i) * (2 ! i). Write thedivision in fraction form. Then multiply numerator and denominator by 2 $ i .


5. How can you use what you know about simplifying an expression such as to

help you remember how to simplify fractions with imaginary numbers in thedenominator? Sample answer: In both cases, you can multiply thenumerator and denominator by the conjugate of the denominator.

1 % #3$&2 ! #5$

2 $ i

3 ! 7i

pure imaginary 70

real03

imaginary54

standard form

1!4!2


Conjugates and Absolute ValueWhen studying complex numbers, it is often convenient to represent a complex number by a single variable. For example, we might let z " x % yi. We denote the conjugate of z by z$. Thus, z$ " x ! yi.

We can define the absolute value of a complex number as follows.

- z- " - x % yi- " #x2 % y$2$

There are many important relationships involving conjugates and absolute values of complex numbers.

Show +z +2 & zz" for any complex number z.

Let z " x % yi. Then,z " (x % yi)(x ! yi)

" x2 % y2

" #(x2 % y2$)2$" - z-2

Show is the multiplicative inverse for any nonzero

complex number z.

We know - z-2 " zz$. If z # 0, then we have z! " " 1.

Thus, is the multiplicative inverse of z.

For each of the following complex numbers, find the absolute value andmultiplicative inverse.

1. 2i 2. !4 ! 3i 3. 12 ! 5i

4. 5 ! 12i 5. 1 % i 6. #3$ ! i

7. % i 8. ! i 9. &12& ! i#3$

&2#2$&2

#2$&2

#3$&3

#3$&3

z$&- z-2

z$&- z-2

z""+z +2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

5-95-9

Example 1Example 1

Example 2Example 2

© Glencoe/McGraw-Hill A2 Glencoe Algebra 2

Answers (Lesson 5-1)

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5-1

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2 #

107

3.21

9 #

1012

3 #

10!

5

25.C

OM

PUTI

NG

The

ter

m b

it,s

hort

for

bin

ary

digi

t,w

as f

irst

use

d in

194

6 by

Joh

n T

ukey

.A

sin

gle

bit

hold

s a

zero

or

a on

e.So

me

com

pute

rs u

se 3

2-bi

t nu

mbe

rs,o

r st

ring

s of

32

con

secu

tive

bit

s,to

iden

tify

eac

h ad

dres

s in

the

ir m

emor

ies.

Eac

h 32

-bit

num

ber

corr

espo

nds

to a

num

ber

in o

ur b

ase-

ten

syst

em.T

he la

rges

t 32

-bit

num

ber

is n

earl

y4,

295,

000,

000.

Wri

te t

his

num

ber

in s

cien

tifi

c no

tati

on.

4.29

5 #

109

26.L

IGH

TW

hen

light

pas

ses

thro

ugh

wat

er,i

ts v

eloc

ity

is r

educ

ed b

y 25

%.I

f th

e sp

eed

oflig

ht in

a v

acuu

m is

1.8

6 '

105

mile

s pe

r se

cond

,at

wha

t ve

loci

ty d

oes

it t

rave

l thr

ough

wat

er?

Wri

te y

our

answ

er in

sci

enti

fic

nota

tion

. 1.3

95 #

105

mi/s

27

.TR

EES

Dec

iduo

us a

nd c

onif

erou

s tr

ees

are

hard

to

dist

ingu

ish

in a

bla

ck-a

nd-w

hite

phot

o.B

ut b

ecau

se d

ecid

uous

tre

es r

efle

ct in

frar

ed e

nerg

y be

tter

tha

n co

nife

rous

tre

es,

the

two

type

s of

tre

es a

re m

ore

dist

ingu

isha

ble

in a

n in

frar

ed p

hoto

.If

an in

frar

edw

avel

engt

h m

easu

res

abou

t 8

'10

!7

met

ers

and

a bl

ue w

avel

engt

h m

easu

res

abou

t 4.

5 '

10!

7m

eter

s,ab

out

how

man

y ti

mes

long

er is

the

infr

ared

wav

elen

gth

than

the

blue

wav

elen

gth?

abo

ut 1

.8 ti

mes

2.7

'10

6&

&9

'10

10

4v2

" m2

!20

(m2 v

)(!

v)3

&&

5(!

v)2 (

!m

4 )15

x11"

y3(3

x!2 y

3 )(5

xy!

8 )&

&(x

!3 )

4 y!

2

y6" 4x

22x

3 y2

& !x2 y

54 & 3

3 & 2

256

" wz5

4" 9r

4 s6 z

122

& 3r2 s

3 z6

27x6

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x7 )&

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s4" 3x

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3& 18

sx7

4m7 y

2"

312

m8 y

6& !

9my4

20d

3 t2

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" b61 " x4

Pra

ctic

e (A

vera

ge)

Mon

omia

ls

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-1

5-1



Rea

din

g t

o L

earn

Math

emati

csM

onom

ials

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-1

5-1

©G

lenc

oe/M

cGra

w-Hi

ll24

3G

lenc

oe A

lgeb

ra 2

Lesson 5-1

Pre-

Act

ivit

yW

hy

is s

cien

tifi

c n

otat

ion

use

ful

in e

con

omic

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

1 at

the

top

of

page

222

in y

our

text

book

.

Your

tex

tboo

k gi

ves

the

U.S

.pub

lic d

ebt

as a

n ex

ampl

e fr

om e

cono

mic

s th

atin

volv

es la

rge

num

bers

tha

t ar

e di

ffic

ult

to w

ork

wit

h w

hen

wri

tten

inst

anda

rd n

otat

ion.

Giv

e an

exa

mpl

e fr

om s

cien

ce t

hat

invo

lves

ver

y la

rge

num

bers

and

one

tha

t in

volv

es v

ery

smal

l num

bers

.Sa

mpl

e an

swer

:di

stan

ces

betw

een

Eart

h an

d th

e st

ars,

size

s of

mol

ecul

es

and

atom

s

Rea

din

g t

he

Less

on

1.Te

ll w

heth

er e

ach

expr

essi

on is

a m

onom

ial o

r no

t a

mon

omia

l.If

it is

a m

onom

ial,

tell

whe

ther

it is

a c

onst

ant

or n

ot a

con

stan

t.

a.3x

2m

onom

ial;

not a

con

stan

tb.

y2%

5y!

6no

t a m

onom

ial

c.!

73m

onom

ial;

cons

tant

d.no

t a m

onom

ial

2.C

ompl

ete

the

follo

win

g de

fini

tion

s of

a n

egat

ive

expo

nent

and

a z

ero

expo

nent

.

For

any

real

num

ber

a#

0 an

d an

y in

tege

r n,

a!n

".

For

any

real

num

ber

a#

0,a0

".

3.N

ame

the

prop

erty

or

prop

erti

es o

f ex

pone

nts

that

you

wou

ld u

se t

o si

mpl

ify

each

expr

essi

on.(

Do

not

actu

ally

sim

plif

y.)

a.qu

otie

nt o

f pow

ers

b.y6

$y9

prod

uct o

f pow

ers

c.(3

r2s)

4po

wer

of a

pro

duct

and

pow

er o

f a p

ower

Hel

pin

g Y

ou

Rem

emb

er

4.W

hen

wri

ting

a n

umbe

r in

sci

enti

fic

nota

tion

,som

e st

uden

ts h

ave

trou

ble

rem

embe

ring

whe

n to

use

pos

itiv

e ex

pone

nts

and

whe

n to

use

neg

ativ

e on

es.W

hat

is a

n ea

sy w

ay t

ore

mem

ber

this

?Sa

mpl

e an

swer

:Use

a p

ositi

ve e

xpon

ent i

f the

num

ber i

s10

or g

reat

er.U

se a

neg

ativ

e nu

mbe

r if t

he n

umbe

r is

less

than

1.

m8

& m3

1

" a1 n"

1 & z

©G

lenc

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cGra

w-Hi

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4G

lenc

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lgeb

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Prop

ertie

s of

Exp

onen

tsT

he r

ules

abo

ut p

ower

s an

d ex

pone

nts

are

usua

lly g

iven

wit

h le

tter

s su

ch a

s m

,n,

and

kto

rep

rese

nt e

xpon

ents

.For

exa

mpl

e,on

e ru

le s

tate

s th

at a

m$

an"

am%

n .

In p

ract

ice,

such

exp

onen

ts a

re h

andl

ed a

s al

gebr

aic

expr

essi

ons

and

the

rule

s of

al

gebr

a ap

ply.

Sim

pli

fy 2

a2 (

an

$1

$a

4n).

2a2 (

an%

1%

a4n )

"2a

2$

an%

1%

2a2

$a4

nUs

e th

e Di

strib

utive

Law

.

"2a

2 %

n%

1%

2a2

%4n

Reca

ll am

$an

"am

%n .

"2a

n%

3%

2a2

%4n

Sim

plify

the

expo

nent

2 %

n%

1 as

n%

3.

It is

impo

rtan

t al

way

s to

col

lect

like

term

s on

ly.

Sim

pli

fy (

an

$bm

)2.

(an

%bm

)2"

(an

%bm

)(an

%bm

)F

OI

L"

an$

an%

an$

bm%

an$

bm%

bm$

bmTh

e se

cond

and

third

term

s ar

e lik

e te

rms.

" a

2n%

2an b

m%

b2m

Sim

pli

fy e

ach

exp

ress

ion

by

per

form

ing

the

ind

icat

ed o

per

atio

ns.

1.23

2m23

$m

2.(a

3 )n

a3n

3.(4

n b2 )

k4k

n b2k

4.(x

3 aj )

mx3

maj

m5.

(!ay

n )3

!a3

y3n

6.(!

bkx)

2!

b2k x

2

7.(c

2 )hk

c2hk

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2dn )

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32d5

n9.

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b2)

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n b3

10.(

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3 !

n

13.(

ab2

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1

14.a

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bn%

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2a3 b

n $

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4a2 b

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12x3

& 4xn

an& a2

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-1

5-1

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Poly

nom

ials

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-2

5-2

©G

lenc

oe/M

cGra

w-Hi

ll24

5G

lenc

oe A

lgeb

ra 2

Lesson 5-2

Ad

d a

nd

Su

btr

act

Poly

no

mia

ls

Poly

nom

ial

a m

onom

ial o

r a s

um o

f mon

omia

ls

Like

Ter

ms

term

s th

at h

ave

the

sam

e va

riabl

e(s)

raise

d to

the

sam

e po

wer(s

)

To a

dd o

r su

btra

ct p

olyn

omia

ls,p

erfo

rm t

he in

dica

ted

oper

atio

ns a

nd c

ombi

ne li

ke t

erm

s.

Sim

pli

fy !

6rs

$18

r2!

5s2

! 1

4r2

$8r

s !

6s2 .

!6r

s%

18r2

!5s

2!

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%8r

s!6s

2

"(1

8r2

!14

r2) %

(!6r

s%

8rs)

%(!

5s2

!6s

2 )G

roup

like

term

s."

4r2

%2r

s!

11s2

Com

bine

like

term

s.

Sim

pli

fy 4

xy2

$12

xy!

7x2 y

!(2

0xy

$5x

y2!

8x2 y

).

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xy!

7x2 y

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)"

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xy!

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!20

xy!

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%8x

2 yDi

strib

ute

the

min

us s

ign.

" (!

7x2 y

%8x

2 y) %

(4xy

2!

5xy2

) %(1

2xy

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xy)

Gro

up li

ke te

rms.

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y!

xy2

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yCo

mbi

ne li

ke te

rms.

Sim

pli

fy.

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x2!

3x%

2) !

(4x2

%x

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7 " 81 " 8

3 & 81 & 4

1 & 21 & 2

3 & 81 & 4Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

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lenc

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cGra

w-Hi

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6G

lenc

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lgeb

ra 2

Mu

ltip

ly P

oly

no

mia

lsYo

u us

e th

e di

stri

buti

ve p

rope

rty

whe

n yo

u m

ulti

ply

poly

nom

ials

.Whe

n m

ulti

plyi

ng b

inom

ials

,the

FO

ILpa

tter

n is

hel

pful

.

To m

ultip

ly tw

o bi

nom

ials,

add

the

prod

ucts

of

Fth

e fir

stte

rms,

FOIL

Pat

tern

Oth

e ou

tert

erm

s,I

the

inne

rter

ms,

and

Lth

e la

stte

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Fin

d 4

y(6

!2y

$5y

2 ).

4y(6

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2 ) "

4y(6

) %

4y(!

2y) %

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y2)

Dist

ribut

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Mul

tiply

the

mon

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ls.

Fin

d (

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x$

1).

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(2x

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%(!

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Firs

t ter

ms

Out

er te

rms

Inne

r ter

ms

Last

term

s

"12

x2%

6x!

10x

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Mul

tiply

mon

omia

ls."

12x2

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Add

like

term

s.

Fin

d e

ach

pro

du

ct.

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2.

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)3.

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2 (y2

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x$

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20.(

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$7x

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Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Poly

nom

ials

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-2

5-2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Poly

nom

ials

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-2

5-2

©G

lenc

oe/M

cGra

w-Hi

ll24

7G

lenc

oe A

lgeb

ra 2

Lesson 5-2

Det

erm

ine

wh

eth

er e

ach

exp

ress

ion

is

a p

olyn

omia

l.If

it

is a

pol

ynom

ial,

stat

e th

ed

egre

e of

th

e p

olyn

omia

l.

1.x2

%2x

%2

yes;

22.

no3.

8xz

%y

yes;

2

Sim

pli

fy.

4.(g

%5)

%(2

g%

7)5.

(5d

%5)

!(d

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3g$

124d

$4

6.(x

2!

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3) %

(2x2

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grow

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NAM

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____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-2

5-2


An

swer

s


Rea

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o L

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Math

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ls

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

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OD

____

_

5-2

5-2

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Lesson 5-2

Pre-

Act

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an p

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s?

Rea

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e in

trod

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on t

o L

esso

n 5-

2 at

the

top

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page

229

in y

our

text

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.

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She

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five

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text

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w c

ould

she

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e th

e tu

itio

n fo

r th

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fth

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prog

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Poly

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En

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NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-2

5-2



Stu

dy

Gu

ide

and I

nte

rven

tion

Divi

ding

Pol

ynom

ials

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-3

5-3

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Lesson 5-3

Use

Lo

ng

Div

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div

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lyno

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a m

onom

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use

the

prop

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es o

f po

wer

sfr

om L

esso

n 5-

1.

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poly

nom

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y a

poly

nom

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use

a lo

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ivis

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patt

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nly

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term

s ca

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Use

Syn

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Synt

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Use

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ivis

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scen

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er. T

hen

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the

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3) *

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1)

3x2

!x

$7

5x2

$2x

!3

3.(2

x3%

3x2

!10

x !

3) *

(x%

3)4.

(x3

!8x

2%

19x

!9)

*(x

!4)

2x2

!3x

!1

x2!

4x$

3 $

5.(2

x3%

10x2

%9x

%38

) *

(x%

5)6.

(3x3

!8x

2%

16x

!1)

*(x

!1)

2x2

$9

!3x

2!

5x$

11 $

7.(x

3!

9x2

%17

x!

1) *

(x!

2)8.

(4x3

!25

x2%

4x%

20)

*(x

!6)

x2!

7x$

3 $

4x2

!x

!2

$

9.(6

x3%

28x2

!7x

%9)

*(x

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10.(

x4!

4x3

%x2

%7x

!2)

*(x

!2)

6x2

!2x

$3

!x3

!2x

2!

3x$

1

11.(

12x4

%20

x3!

24x2

%20

x%

35)

*(3

x%

5)4x

3!

8x$

20 $

!65

" 3x$

5

6" x

$5

8" x

!6

5" x

!2

10" x

!1

7" x

$5

3" x

!4

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Divi

ding

Pol

ynom

ials

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-3

5-3

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Divi

ding

Pol

ynom

ials

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-3

5-3

©G

lenc

oe/M

cGra

w-Hi

ll25

3G

lenc

oe A

lgeb

ra 2

Lesson 5-3

Sim

pli

fy.

1.5c

$3

2.3x

$5

3.5y

2$

2y$

14.

3x!

1 !

5.(1

5q6

%5q

2 )(5

q4)!

16.

(4f5

!6f

4%

12f3

!8f

2 )(4

f2)!

1

3q2

$f3

!$

3f!

2

7.(6

j2k

!9j

k2) *

3jk

8.(4

a2h2

!8a

3 h%

3a4 )

*(2

a2)

2j!

3k2h

2!

4ah

$

9.(n

2%

7n%

10)

*(n

%5)

10.(

d2

%4d

%3)

*(d

%1)

n$

2d

$3

11.(

2s2

%13

s%

15)

*(s

%5)

12.(

6y2

%y

!2)

(2y

!1)

!1

2s$

33y

$2

13.(

4g2

!9)

*(2

g%

3)14

.(2x

2!

5x!

4) *

(x!

3)

2g!

32x

$1

!

15.

16.

u$

8 $

2x$

1 !

17.(

3v2

!7v

!10

)(v

!4)

!1

18.(

3t4

%4t

3!

32t2

!5t

!20

)(t

%4)

!1

3v$

5 $

3t3

!8t

2!

5

19.

20.

y2!

3y$

6 !

2x2

$5x

!4

$

21.(

4p3

!3p

2%

2p) *

(p!

1)22

.(3c

4%

6c3

!2c

%4)

(c%

2)!

1

4p2

$p

$3

$3c

3!

2 $

23.G

EOM

ETRY

The

are

a of

a r

ecta

ngle

is x

3%

8x2

%13

x!

12 s

quar

e un

its.

The

wid

th o

fth

e re

ctan

gle

is x

%4

unit

s.W

hat

is t

he le

ngth

of

the

rect

angl

e?x2

$4x

!3

units

8" c

$2

3" p

!1

3" x

!3

18" y

$2

2x3

!x2

!19

x%

15&

&&

x!

3y3

!y2

!6

&&

y%

2

10" v

!4

1" x

!3

12" u

!3

2x2

!5x

!4

&&

x!

3u2

%5u

!12

&&

u!

3

1" x

!33a

2"

2

3f2

"2

1 " q2

2 " x12

x2!

4x!

8&

& 4x15

y3%

6y2

%3y

&& 3y

12x

%20

&& 4

10c

%6

&2

©G

lenc

oe/M

cGra

w-Hi

ll25

4G

lenc

oe A

lgeb

ra 2

Sim

pli

fy.

1.3r

6!

r4$

2.!

6k2

$

3.(!

30x3

y%

12x2

y2!

18x2

y) *

(!6x

2 y)

4.(!

6w3 z

4!

3w2 z

5%

4w%

5z) *

(2w

2 z)

5x!

2y$

3!

3wz3

!$

$

5.(4

a3!

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%a2

)(4a

)!1

6.(2

8d3 k

2%

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2!

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)(4d

k2)!

1

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"a 4"7d

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1

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9.(a

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64)

*(a

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a2$

4a$

1610

.(b3

%27

) *

(b%

3)b2

!3b

$9

11.

2x2

!8x

$38

12.

2x2

!6x

$22

!

13.(

3w3

%7w

2!

4w%

3) *

(w%

3)14

.(6y

4%

15y3

!28

y!

6) *

(y%

2)

3w2

!2w

$2

!" w

$33

"6y

3$

3y2

!6y

!16

$" y2 $6 2

"

15.(

x4!

3x3

!11

x2%

3x%

10)

*(x

!5)

16.(

3m5

%m

!1)

*(m

%1)

x3$

2x2

!x

!2

3m4

!3m

3$

3m2

!3m

$4

!

17.(

x4!

3x3

%5x

!6)

(x%

2)!

118

.(6y

2!

5y!

15)(

2y%

3)!

1

x3!

5x2

$10

x!

15 $

" x2 $4 2"

3y!

7 $

" 2y6 $

3"

19.

20.

2x$

2 $

" 2x1 !2

3"

2x!

1 !

" 3x6 $

1"

21.(

2r3

%5r

2!

2r!

15)

*(2

r!

3)22

.(6t

3%

5t2

!2t

%1)

*(3

t%

1)r2

$4r

$5

2t2

$t!

1 $

" 3t2 $

1"

23.

24.

2p3

$3p

2!

4p$

12h

2!

h$

325

.GEO

MET

RYT

he a

rea

of a

rec

tang

le is

2x2

!11

x%

15 s

quar

e fe

et.T

he le

ngth

of

the

rect

angl

e is

2x

!5

feet

.Wha

t is

the

wid

th o

f th

e re

ctan

gle?

x!

3 ft

26.G

EOM

ETRY

The

are

a of

a t

rian

gle

is 1

5x4

%3x

3%

4x2

!x

!3

squa

re m

eter

s.T

hele

ngth

of

the

base

of

the

tria

ngle

is 6

x2!

2 m

eter

s.W

hat

is t

he h

eigh

t of

the

tri

angl

e?5x

2$

x$

3 m

2h4

!h3

%h2

%h

!3

&&

&h2

!1

4p4

!17

p2%

14p

!3

&&

&2p

!3

6x2

!x

! 7

&&

3x%

14x

2!

2x%

6&

&2x

!3

5" m

$1

72" x

$3

2x3

%4x

!6

&&

x%

32x

3%

6x%

152

&&

x%

4

2x2

%3x

!14

&&

x!

2f2

%7f

%10

&&

f%

2

5" 2w

22 " wz

3z4

"2

9m " 2k3k " m

6k2 m

!12

k3 m2

%9m

3&

&&

2km

28 " r2

15r10

!5r

8%

40r2

&&

&5r

4Pra

ctic

e (A

vera

ge)

Divi

ding

Pol

ynom

ials

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-3

5-3



Rea

din

g t

o L

earn

Math

emati

csDi

vidi

ng P

olyn

omia

ls

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-3

5-3

©G

lenc

oe/M

cGra

w-Hi

ll25

5G

lenc

oe A

lgeb

ra 2

Lesson 5-3

Pre-

Act

ivit

yH

ow c

an y

ou u

se d

ivis

ion

of

pol

ynom

ials

in

man

ufa

ctu

rin

g?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

3 at

the

top

of

page

233

in y

our

text

book

.

Usi

ng t

he d

ivis

ion

sym

bol (

*),

wri

te t

he d

ivis

ion

prob

lem

tha

t yo

u w

ould

use

to a

nsw

er t

he q

uest

ion

aske

d in

the

intr

oduc

tion

.(D

o no

t ac

tual

lydi

vide

.)(3

2x2

$x)

%(8

x)

Rea

din

g t

he

Less

on

1.a.

Exp

lain

in w

ords

how

to

divi

de a

pol

ynom

ial b

y a

mon

omia

l.Di

vide

eac

h te

rm o

fth

e po

lyno

mia

l by

the

mon

omia

l.b.

If y

ou d

ivid

e a

trin

omia

l by

a m

onom

ial a

nd g

et a

pol

ynom

ial,

wha

t ki

nd o

fpo

lyno

mia

l will

the

quo

tien

t be

?tri

nom

ial

2.L

ook

at t

he f

ollo

win

g di

visi

on e

xam

ple

that

use

s th

e di

visi

on a

lgor

ithm

for

pol

ynom

ials

.

2x%

4x

!4%$

$2x

2!

4x%

72x

2!

8x 4x%

74x

!16 23

Whi

ch o

f th

e fo

llow

ing

is t

he c

orre

ct w

ay t

o w

rite

the

quo

tien

t?C

A.

2x%

4B

.x

!4

C.

2x%

4 %

D.

3.If

you

use

syn

thet

ic d

ivis

ion

to d

ivid

e x3

%3x

2!

5x!

8 by

x!

2,th

e di

visi

on w

ill lo

oklik

e th

is:

21

3!

5!

82

1010

15

52

Whi

ch o

f th

e fo

llow

ing

is t

he a

nsw

er f

or t

his

divi

sion

pro

blem

?B

A.

x2%

5x%

5B

.x2

%5x

%5

%

C.

x3%

5x2

%5x

%D

.x3

%5x

2%

5x%

2

Hel

pin

g Y

ou

Rem

emb

er

4.W

hen

you

tran

slat

e th

e nu

mbe

rs in

the

last

row

of

a sy

nthe

tic

divi

sion

into

the

quo

tien

tan

d re

mai

nder

,wha

t is

an

easy

way

to

rem

embe

r w

hich

exp

onen

ts t

o us

e in

wri

ting

the

term

s of

the

quo

tien

t?Sa

mpl

e an

swer

:Sta

rt w

ith th

e po

wer

that

is o

ne le

ssth

an th

e de

gree

of t

he d

ivid

end.

Decr

ease

the

pow

er b

y on

e fo

r eac

hte

rm a

fter t

he fi

rst.

The

final

num

ber w

ill b

e th

e re

mai

nder

.Dro

p an

y te

rmth

at is

repr

esen

ted

by a

0.

2& x

!2

2& x

!2

23& x

!4

23& x

!4

©G

lenc

oe/M

cGra

w-Hi

ll25

6G

lenc

oe A

lgeb

ra 2

Obl

ique

Asy

mpt

otes

The

gra

ph o

f y"

ax%

b,w

here

a #

0,i

s ca

lled

an o

bliq

ue a

sym

ptot

e of

y"

f(x)

if

the

gra

ph o

f fco

mes

clo

ser

and

clos

er t

o th

e lin

e as

x →

∞or

x →

!∞

.∞is

the

mat

hem

atic

al s

ymbo

l for

in

fin

ity,

whi

ch m

eans

end

less

.

For

f(x)

"3x

%4

%&2 x& ,

y"

3x%

4 is

an

obliq

ue a

sym

ptot

e be

caus

e

f(x)

!3x

!4

"&2 x& ,

and

&2 x&→

0 a

s x →

∞or

!∞

.In

othe

r w

ords

,as

|x|

incr

ease

s,th

e va

lue

of &2 x&

gets

sm

alle

r an

d sm

alle

r ap

proa

chin

g 0.

Fin

d t

he

obli

que

asym

pto

te f

or f

(x)

&"x2

$ x8 $x

2$15

".

!2

18

15Us

e sy

nthe

tic d

ivisio

n.!

2!

121

63

y"&x2

% x8 %x2%

15&

"x

%6

%& x

%32

&

As

|x|i

ncre

ases

,the

val

ue o

f &x

%32

&ge

ts s

mal

ler.

In o

ther

wor

ds,s

ince

& x%3

2&

→ 0

as

x →

∞or

x →

!∞

,y"

x%

6 is

an

obliq

ue a

sym

ptot

e.

Use

syn

thet

ic d

ivis

ion

to

fin

d t

he

obli

que

asym

pto

te f

or e

ach

fu

nct

ion

.

1.y

"&8x

2! x

%4x5%

11&

y&

8x!

44

2.y

"&x2

% x3 !x2!

15&

y&

x$

5

3.y

"&x2

! x2 !x3!

18&

y&

x$

1

4.y

"&ax

2

x%!bx

d%

c&

y&

ax$

b$

ad

5.y

"&ax

2

x%%bx

d%

c&

y&

ax$

b!

ad

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-3

5-3

Exam

ple

Exam

ple


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Fact

orin

g Po

lyno

mia

ls

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-4

5-4

©G

lenc

oe/M

cGra

w-Hi

ll25

7G

lenc

oe A

lgeb

ra 2

Lesson 5-4

Fact

or

Poly

no

mia

ls

For a

ny n

umbe

r of t

erm

s, c

heck

for:

grea

test

com

mon

fact

or

For t

wo te

rms,

che

ck fo

r:Di

ffere

nce

of tw

o sq

uare

sa2

!b2

"(a

%b)

(a!

b)Su

m o

f two

cub

esa3

%b3

"(a

%b)

(a2

!ab

%b2

)Di

ffere

nce

of tw

o cu

bes

a3!

b3"

(a!

b)(a

2%

ab%

b2)

Tech

niqu

es fo

r Fac

torin

g Po

lyno

mia

lsFo

r thr

ee te

rms,

che

ck fo

r:Pe

rfect

squ

are

trino

mia

lsa2

%2a

b%

b2"

(a%

b)2

a2!

2ab

%b2

"(a

!b)

2

Gen

eral

trin

omia

lsac

x2%

(ad

%bc

)x%

bd"

(ax

%b)

(cx

%d)

For f

our t

erm

s, c

heck

for:

Gro

upin

gax

%bx

%ay

%by

"x(

a%

b) %

y(a

%b)

"(a

%b)

(x%

y)

Fact

or 2

4x2

!42

x!

45.

Fir

st f

acto

r ou

t th

e G

CF

to

get

24x2

!42

x!

45 "

3(8x

2!

14x

!15

).To

fin

d th

e co

effi

cien

tsof

the

xte

rms,

you

mus

t fi

nd t

wo

num

bers

who

se p

rodu

ct is

8 $

(!15

) "

!12

0 an

d w

hose

sum

is !

14.T

he t

wo

coef

fici

ents

mus

t be

!20

and

6.R

ewri

te t

he e

xpre

ssio

n us

ing

!20

xan

d6x

and

fact

or b

y gr

oupi

ng.

8x2

!14

x!

15 "

8x2

!20

x%

6x!

15G

roup

to fi

nd a

GCF

."

4x(2

x!

5) %

3(2x

!5)

Fact

or th

e G

CF o

f eac

h bi

nom

ial.

"(4

x%

3)(2

x!

5)Di

strib

utive

Pro

perty

Thu

s,24

x2!

42x

!45

"3(

4x%

3)(2

x!

5).

Fact

or c

omp

lete

ly.I

f th

e p

olyn

omia

l is

not

fac

tora

ble,

wri

te p

rim

e.

1.14

x2y2

%42

xy3

2.6m

n%

18m

!n

!3

3.2x

2%

18x

%16

14xy

2 (x

$3y

)(6

m!

1)(n

$3)

2(

x$

8)(x

$1)

4.x4

!1

5.35

x3y4

!60

x4y

6.2r

3%

250

(x2

$1)

(x$

1)(x

!1)

5x

3 y(7

y3!

12x)

2(r$

5)(r

2!

5r$

25)

7.10

0m8

!9

8.x2

%x

%1

9.c4

%c3

!c2

!c

(10m

4!

3)(1

0m4

$3)

prim

ec(

c$

1)2 (

c!

1)

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-Hi

ll25

8G

lenc

oe A

lgeb

ra 2

Sim

plif

y Q

uo

tien

tsIn

the

last

less

on y

ou le

arne

d ho

w t

o si

mpl

ify

the

quot

ient

of

two

poly

nom

ials

by

usin

g lo

ng d

ivis

ion

or s

ynth

etic

div

isio

n.So

me

quot

ient

s ca

n be

sim

plif

ied

byus

ing

fact

orin

g.

Sim

pli

fy

.

"Fa

ctor

the

num

erat

or a

nd d

enom

inat

or.

"Di

vide.

Ass

ume

x#

8, !

.

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o 0.

1.2.

3.

4.5.

6.

7.8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

9x2

$6x

$4

""

3x$

2y2

$4y

$16

""

3y!

52x

!3

""

4x2

!2x

$1

27x3

!8

&&

9x2

!4

y3!

64&

&3y

2!

17y

%20

4x2

!4x

!3

&&

8x3

%1

2x!

3" x

$8

7x!

3" x

!2

x$

9" 2x

!5

4x2

!12

x%

9&

&2x

2%

13x

!24

7x2

%11

x!

6&

&x2

!4

x2!

81&

&2x

2!

23x

%45

x!

7" x

$5

3x!

5" 2x

!3

2x$

5" x

!1

x2!

14x

%49

&&

x2!

2x!

353x

2%

4x!

15&

&2x

2%

3x!

94x

2%

16x

%15

&&

2x2

%x

!3

x!

2" 3x

$7

6x$

1" x

$2

2x!

1" x

!2

x2!

7x%

10&

&3x

2!

8x!

356x

2%

25x

%4

&&

x2%

6x%

84x

2%

4x!

3&

&2x

2!

x!

6

5x!

1" x

$3

2x!

1" 4x

!1

x$

3" x

!5

5x2

%9x

!2

&&

x2%

5x%

62x

2%

5x!

3&

&4x

2%

11x

!3

x2%

x!

6&

&x2

!7x

%10

x!

5" x

$1

x$

5" 2x

!3

x!

4" x

$2

x2!

11x

%30

&&

x2!

5x!

6x2

%6x

%5

&&

2x2

!x

!3

x2!

7x%

12&

&x2

!x

!6

3 & 2x

%4

& x!

8

(2x

%3)

( x%

4)&

&(x

!8)

(2x

%3)

8x2

%11

x%

12&

&2x

2!

13x

!24

8x2

%11

x%

12"

"2x

2!

13x

!24

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Fact

orin

g Po

lyno

mia

ls

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-4

5-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Fact

orin

g Po

lyno

mia

ls

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-4

5-4

©G

lenc

oe/M

cGra

w-Hi

ll25

9G

lenc

oe A

lgeb

ra 2

Lesson 5-4

Fact

or c

omp

lete

ly.I

f th

e p

olyn

omia

l is

not

fac

tora

ble,

wri

te p

rim

e.

1.7x

2!

14x

2.19

x3!

38x2

7x(x

!2)

19x2

(x!

2)

3.21

x3!

18x2

y%

24xy

24.

8j3 k

!4j

k3!

73x

(7x2

!6x

y$

8y2 )

prim

e

5.a2

%7a

!18

6.2a

k!

6a%

k!

3(a

$9)

(a!

2)(2

a$

1)(k

!3)

7.b2

%8b

%7

8.z2

!8z

!10

(b$

7)(b

$1)

prim

e

9.m

2%

7m!

1810

.2x2

!3x

!5

(m!

2)(m

$9)

(2x

!5)

(x$

1)

11.4

z2%

4z!

1512

.4p2

%4p

!24

(2z

$5)

(2z

!3)

4(p

!2)

(p$

3)

13.3

y2%

21y

%36

14.c

2!

100

3(y

$4)

(y$

3)(c

$10

)(c!

10)

15.4

f2!

6416

.d2

!12

d%

364(

f$4)

(f!

4)(d

!6)

2

17.9

x2%

2518

.y2

%18

y%

81pr

ime

(y$

9)2

19.n

3!

125

20.m

4!

1(n

!5)

(n2

$5n

$25

)(m

2$

1)(m

!1)

(m$

1)

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o 0.

21.

22.

23.

24.

x!

1" x

!7

x2%

6x!

7&

&x2

!49

x!

5"

xx2

!10

x%

25&

&x2

!5x

x$

1" x

$3

x2%

4x%

3&

&x2

%6x

%9

x!

2" x

!5

x2%

7x!

18&

&x2

%4x

!45

©G

lenc

oe/M

cGra

w-Hi

ll26

0G

lenc

oe A

lgeb

ra 2

Fact

or c

omp

lete

ly.I

f th

e p

olyn

omia

l is

not

fac

tora

ble,

wri

te p

rim

e.

1.15

a2b

!10

ab2

2.3s

t2!

9s3 t

%6s

2 t2

3.3x

3 y2

!2x

2 y%

5xy

5ab(

3a!

2b)

3st(

t!3s

2$

2st)

xy(3

x2y

!2x

$5)

4.2x

3 y!

x2y

%5x

y2%

xy3

5.21

!7t

%3r

!rt

6.x2

!xy

%2x

!2y

xy(2

x2!

x$

5y $

y2)

(7 $

r)(3

!t)

(x$

2)(x

!y)

7.y2

%20

y%

968.

4ab

%2a

%6b

%3

9.6n

2!

11n

!2

(y$

8)(y

$12

)(2

a$

3)(2

b$

1)(6

n$

1)(n

!2)

10.6

x2%

7x!

311

.x2

!8x

!8

12.6

p2!

17p

!45

(3x

!1)

(2x

$3)

prim

e(2

p!

9)(3

p$

5)13

.r3

%3r

2!

54r

14.8

a2%

2a!

615

.c2

!49

r(r$

9)(r

!6)

2(4a

!3)

(a$

1)(c

!7)

(c$

7)16

.x3

%8

17.1

6r2

!16

918

.b4

!81

(x$

2)(x

2!

2x$

4)(4

r$13

)(4r!

13)

(b2

$9)

(b$

3)(b

!3)

19.8

m3

!25

prim

e20

.2t3

%32

t2%

128t

2t(t

$8)

2

21.5

y5%

135y

25y

2 (y

$3)

(y2

!3y

$9)

22.8

1x4

!16

(9x2

$4)

(3x

$2)

(3x

!2)

Sim

pli

fy.A

ssu

me

that

no

den

omin

ator

is

equ

al t

o 0.

23.

24.

25.

26.D

ESIG

NB

obbi

Jo

is u

sing

a s

oftw

are

pack

age

to c

reat

e a

draw

ing

of a

cro

ss s

ecti

on o

f a

brac

e as

sho

wn

at t

he r

ight

.W

rite

a s

impl

ifie

d,fa

ctor

ed e

xpre

ssio

n th

at r

epre

sent

s th

e ar

ea o

f th

e cr

oss

sect

ion

of t

he b

race

.x(

20.2

!x)

cm

2

27.C

OM

BU

STIO

N E

NG

INES

In a

n in

tern

al c

ombu

stio

n en

gine

,the

up

and

dow

n m

otio

n of

the

pis

tons

is c

onve

rted

into

the

rot

ary

mot

ion

of

the

cran

ksha

ft,w

hich

dri

ves

the

flyw

heel

.Let

r1

repr

esen

t th

e ra

dius

of

the

fly

whe

el a

t th

e ri

ght

and

let

r 2re

pres

ent

the

radi

us o

f th

ecr

anks

haft

pas

sing

thr

ough

it.I

f th

e fo

rmul

a fo

r th

e ar

ea o

f a

circ

le

is A

"+

r2,w

rite

a s

impl

ifie

d,fa

ctor

ed e

xpre

ssio

n fo

r th

e ar

ea o

f th

e cr

oss

sect

ion

of t

he f

lyw

heel

out

side

the

cra

nksh

aft.

'(r 1

!r 2

)(r1

$r 2

)

r 1r 2

x cm

x cm

8.2

cm

12 cm

3(x

$3)

""

x2$

3x$

93x

2!

27&

&x3

!27

x!

8" x

$9

x2!

16x

%64

&&

x2%

x!

72x

$4

" x$

5x2

!16

&&

x2%

x!

20Pra

ctic

e (A

vera

ge)

Fact

orin

g Po

lyno

mia

ls

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-4

5-4


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csFa

ctor

ing

Poly

nom

ials

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-4

5-4

©G

lenc

oe/M

cGra

w-Hi

ll26

1G

lenc

oe A

lgeb

ra 2

Lesson 5-4

Pre-

Act

ivit

yH

ow d

oes

fact

orin

g ap

ply

to

geom

etry

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

4 at

the

top

of

page

239

in y

our

text

book

.

If a

tri

nom

ial t

hat

repr

esen

ts t

he a

rea

of a

rec

tang

le is

fac

tore

d in

to t

wo

bino

mia

ls,w

hat

mig

ht t

he t

wo

bino

mia

ls r

epre

sent

?th

e le

ngth

and

wid

th o

f the

rect

angl

e

Rea

din

g t

he

Less

on

1.N

ame

thre

e ty

pes

of b

inom

ials

tha

t it

is a

lway

s po

ssib

le t

o fa

ctor

.di

ffere

nce

of tw

osq

uare

s,su

m o

f tw

o cu

bes,

diffe

renc

e of

two

cube

s

2.N

ame

a ty

pe o

f tr

inom

ial t

hat

it is

alw

ays

poss

ible

to

fact

or.

perfe

ct s

quar

etri

nom

ial

3.C

ompl

ete:

Sinc

e x2

%y2

cann

ot b

e fa

ctor

ed,i

t is

an

exam

ple

of a

po

lyno

mia

l.

4.O

n an

alg

ebra

qui

z,M

arle

ne n

eede

d to

fac

tor

2x2

!4x

!70

.She

wro

te t

he f

ollo

win

gan

swer

:(x

%5)

(2x

!14

).W

hen

she

got

her

quiz

bac

k,M

arle

ne f

ound

tha

t sh

e di

d no

tge

t fu

ll cr

edit

for

her

ans

wer

.She

tho

ught

she

sho

uld

have

got

ten

full

cred

it b

ecau

se s

hech

ecke

d he

r w

ork

by m

ulti

plic

atio

n an

d sh

owed

tha

t (x

%5)

(2x

!14

) "

2x2

!4x

!70

.

a.If

you

wer

e M

arle

ne’s

tea

cher

,how

wou

ld y

ou e

xpla

in t

o he

r th

at h

er a

nsw

er w

as n

oten

tire

ly c

orre

ct?

Sam

ple

answ

er:W

hen

you

are

aske

d to

fact

or a

poly

nom

ial,

you

mus

t fac

tor i

t com

plet

ely.

The

fact

oriz

atio

n w

as n

otco

mpl

ete,

beca

use

2x!

14 c

an b

e fa

ctor

ed fu

rthe

r as

2(x

!7)

.

b.W

hat

advi

ce c

ould

Mar

lene

’s t

each

er g

ive

her

to a

void

mak

ing

the

sam

e ki

nd o

f er

ror

in f

acto

ring

in t

he f

utur

e?Sa

mpl

e an

swer

:Alw

ays

look

for a

com

mon

fact

or fi

rst.

If th

ere

is a

com

mon

fact

or,f

acto

r it o

ut fi

rst,

and

then

see

if yo

u ca

n fa

ctor

furt

her.

Hel

pin

g Y

ou

Rem

emb

er

5.So

me

stud

ents

hav

e tr

oubl

e re

mem

beri

ng t

he c

orre

ct s

igns

in t

he f

orm

ulas

for

the

sum

and

diff

eren

ce o

f tw

o cu

bes.

Wha

t is

an

easy

way

to

rem

embe

r th

e co

rrec

t si

gns?

Sam

ple

answ

er:I

n th

e bi

nom

ial f

acto

r,th

e op

erat

ion

sign

is th

e sa

me

asin

the

expr

essi

on th

at is

bei

ng fa

ctor

ed.I

n th

e tri

nom

ial f

acto

r,th

eop

erat

ion

sign

bef

ore

the

mid

dle

term

is th

e op

posi

te o

f the

sig

n in

the

expr

essi

on th

at is

bei

ng fa

ctor

ed.T

he s

ign

befo

re th

e la

st te

rm is

alw

ays

a pl

us.

prim

e

©G

lenc

oe/M

cGra

w-Hi

ll26

2G

lenc

oe A

lgeb

ra 2

Usin

g Pa

ttern

s to

Fac

tor

Stud

y th

e pa

tter

ns b

elow

for

fac

tori

ng t

he s

um a

nd t

he d

iffe

renc

e of

cub

es.

a3%

b3"

(a%

b)(a

2!

ab%

b2)

a3!

b3"

(a!

b)(a

2%

ab%

b2)

Thi

s pa

tter

n ca

n be

ext

ende

d to

oth

er o

dd p

ower

s.St

udy

thes

e ex

ampl

es.

Fact

or a

5$

b5.

Ext

end

the

firs

t pa

tter

n to

obt

ain

a5%

b5"

(a%

b)(a

4!

a3b

%a2

b2!

ab3

%b4

).C

hec

k:(

a%

b)(a

4!

a3b

%a2

b2!

ab3

%b4

) "a5

!a4

b%

a3b2

!a2

b3%

ab4

%a4

b!

a3b2

%a2

b3!

ab4

%b5

"a5

%b5

Fact

or a

5!

b5.

Ext

end

the

seco

nd p

atte

rn t

o ob

tain

a5

!b5

"(a

!b)

(a4

%a3

b%

a2b2

%ab

3%

b4).

Ch

eck

:(a

!b)

(a4

%a3

b%

a2b2

%ab

3%

b4) "

a5%

a4b

%a3

b2%

a2b3

%ab

4

!a4

b!

a3b2

!a2

b3!

ab4

!b5

"a5

!b5

In g

ener

al,i

f nis

an

odd

inte

ger,

whe

n yo

u fa

ctor

an

%bn

or a

n!

bn,o

ne f

acto

r w

ill b

eei

ther

(a%

b) o

r (a

!b)

,dep

endi

ng o

n th

e si

gn o

f th

e or

igin

al e

xpre

ssio

n.T

he o

ther

fac

tor

will

hav

e th

e fo

llow

ing

prop

erti

es:

•T

he f

irst

ter

m w

ill b

e an

!1

and

the

last

ter

m w

ill b

e bn

!1 .

•T

he e

xpon

ents

of a

will

dec

reas

e by

1 a

s yo

u go

fro

m le

ft t

o ri

ght.

•T

he e

xpon

ents

of b

will

incr

ease

by

1 as

you

go

from

left

to

righ

t.•

The

deg

ree

of e

ach

term

will

be

n!

1.•

If t

he o

rigi

nal e

xpre

ssio

n w

as a

n%

bn,t

he t

erm

s w

ill a

lter

nate

ly h

ave

%an

d !

sign

s.•

If t

he o

rigi

nal e

xpre

ssio

n w

as a

n!

bn,t

he t

erm

s w

ill a

ll ha

ve %

sign

s.

Use

th

e p

atte

rns

abov

e to

fac

tor

each

exp

ress

ion

.

1.a7

%b7

(a$

b)(a

6!

a5b

$a4

b2!

a3b3

$a2

b4!

ab5

$b6

)2.

c9!

d9

(c!

d)(c

8$

c7d

$c6

d2

$c5

d3$

c4d4

$c3

d5$

c2d

6$

cd7

$d8

)3.

e11

%f1

1

(e$

f)(e

10!

e9f$

e8f2

!e7

f3$

e6f4

!e5

f5$

e4f6

!e3

f7$

e2f8

!ef

9$

f10 )

To f

acto

r x1

0!

y10 ,

chan

ge i

t to

(x5

$y5

)(x5

!y5

) an

d f

acto

r ea

ch b

inom

ial.

Use

this

ap

pro

ach

to

fact

or e

ach

exp

ress

ion

.

4.x1

0!

y10

(x$

y)(x

4!

x3y

$x2

y2!

xy3

$y4

)(x!

y)(x

4$

x3y

$x2

y2$

xy3

$y4

)5.

a14

!b1

4(a

$b)

(a6

!a5

b$

a4b2

!a3

b3$

a2b4

!ab

5$

b6)(a

!b)

(a6

$a5

b$

a4b2

$a3

b3$

a2b4

$ab

5$

b6)

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-4

5-4

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2



Stu

dy

Gu

ide

and I

nte

rven

tion

Root

s of

Rea

l Num

bers

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-5

5-5

©G

lenc

oe/M

cGra

w-Hi

ll26

3G

lenc

oe A

lgeb

ra 2

Lesson 5-5

Sim

plif

y R

adic

als

Squa

re R

oot

For a

ny re

al n

umbe

rs a

and

b, if

a2

"b,

then

ais

a sq

uare

root

of b

.

nth

Root

For a

ny re

al n

umbe

rs a

and

b, a

nd a

ny p

ositiv

e in

tege

r n, i

f an

"b,

then

ais

an n

thro

ot o

f b.

1.If

nis

even

and

b,

0, th

en b

has

one

posit

ive ro

ot a

nd o

ne n

egat

ive ro

ot.

Real

nth

Roo

ts o

f b,

2.If

nis

odd

and

b,

0, th

en b

has

one

posit

ive ro

ot.

n !b",

!n !

b"3.

If n

is ev

en a

nd b

)0,

then

bha

s no

real

root

s.4.

If n

is od

d an

d b

)0,

then

bha

s on

e ne

gativ

e ro

ot.

Sim

pli

fy !

49z8

".

#49

z8$

"#

(7z4

)2$

"7z

4

z4m

ust

be p

osit

ive,

so t

here

is n

o ne

ed t

ota

ke t

he a

bsol

ute

valu

e.

Sim

pli

fy !

3 !(2

a!

"1)

6"

!3 #

(2a

!$

1)6

$"

!3 #

[(2a

!$

1)2 ]

3$

"(2

a!

1)2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sim

pli

fy.

1.#

81$2.

3 #!

343

$3.

#14

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$9

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12| p

3 |4.

-#

4a10

$5.

5 #24

3p1

$0 $

6.!

3 #m

6 n9

$(

2a5

3p2

!m

2 n3

7.3 #

!b1

2$

8.#

16a1

0$

b8 $9.

#12

1x6

$!

b44| a

5 | b4

11| x

3 |10

.#(4

k)4

$11

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4$

12.!

3 #!

27p

$6 $

16k2

(13

r23p

2

13.!

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14.#

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2$

y4z6

$!

25| y

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7 |10

| x| y

2 | z3 |

16.

3 #!

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$7$

17.!

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$18

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64p

$10 $

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t a re

al n

umbe

r0.

8|p5

|19

.4 #

(2x)

8$

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(11y

2 )$

4 $21

.3 #

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b)$

6 $4x

212

1y4

25a4

b2

22.#

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!$

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$23

.3 #

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$5)

6$

24.#

36x2

!$

12x

%$

1$| 3

x!

1|(m

!5)

2| 6x

!1|

©G

lenc

oe/M

cGra

w-Hi

ll26

4G

lenc

oe A

lgeb

ra 2

Ap

pro

xim

ate

Rad

ical

s w

ith

a C

alcu

lato

r

Irrat

iona

l Num

ber

a nu

mbe

r tha

t can

not b

e ex

pres

sed

as a

term

inat

ing

or a

repe

atin

g de

cimal

Rad

ical

s su

ch a

s #

2$an

d #

3$ar

e ex

ampl

es o

f ir

rati

onal

num

bers

.Dec

imal

app

roxi

mat

ions

for

irra

tion

al n

umbe

rs a

re o

ften

use

d in

app

licat

ions

.The

se a

ppro

xim

atio

ns c

an b

e ea

sily

foun

d w

ith

a ca

lcul

ator

.

Ap

pro

xim

ate

5 !18

.2"

wit

h a

cal

cula

tor.

5 #18

.2$

&1.

787

Use

a c

alcu

lato

r to

ap

pro

xim

ate

each

val

ue

to t

hre

e d

ecim

al p

lace

s.

1.#

62$2.

#10

50$

3.3 #

0.05

4$

7.87

432

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0.37

8

4.!

4 #5.

45$

5.#

5280

$6.

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,60

$0$

!1.

528

72.6

6413

6.38

2

7.#

0.09

5$

8.3 #

!15

$9.

5 #10

0$

0.30

8!

2.46

62.

512

10.

6 #85

6$

11.#

3200

$12

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05$

3.08

156

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0.22

4

13.#

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0$

0$14

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60$

15.!

4 #50

0$

111.

803

0.77

5!

4.72

9

16.

3 #0.

15$

17.

6 #42

00$

18.#

75$0.

531

4.01

78.

660

19.L

AW

EN

FOR

CEM

ENT

The

for

mul

a r

"2#

5L$is

use

d by

pol

ice

to e

stim

ate

the

spee

d r

in m

iles

per

hour

of

a ca

r if

the

leng

th L

of t

he c

ar’s

ski

d m

ark

is m

easu

res

in f

eet.

Est

imat

e to

the

nea

rest

ten

th o

f a

mile

per

hou

r th

e sp

eed

of a

car

tha

t le

aves

a s

kid

mar

k 30

0 fe

et lo

ng.

77.5

mi/h

20.S

PAC

E TR

AV

ELT

he d

ista

nce

to t

he h

oriz

on d

mile

s fr

om a

sat

ellit

e or

biti

ng h

mile

sab

ove

Ear

th c

an b

e ap

prox

imat

ed b

y d

"#

8000

h$

%h2

$.W

hat

is t

he d

ista

nce

to t

heho

rizo

n if

a s

atel

lite

is o

rbit

ing

150

mile

s ab

ove

Ear

th?

abou

t 110

0 ft

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Root

s of

Rea

l Num

bers

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-5

5-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Root

s of

Rea

l Num

bers

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-5

5-5

©G

lenc

oe/M

cGra

w-Hi

ll26

5G

lenc

oe A

lgeb

ra 2

Lesson 5-5

Use

a c

alcu

lato

r to

ap

pro

xim

ate

each

val

ue

to t

hre

e d

ecim

al p

lace

s.

1.#

230

$15

.166

2.#

38$6.

164

3.!

#15

2$

!12

.329

4.#

5.6

$2.

366

5.3 #

88$4.

448

6.3 #

!22

2$

!6.

055

7.!

4 #0.

34$

!0.

764

8.5 #

500

$3.

466

Sim

pli

fy.

9.-

#81$

(9

10.#

144

$12

11.#

(!5)

2$

512

.#!

52$

not a

real

num

ber

13.#

0.36

$0.

614

.!'(

!

15.

3 #!

8$

!2

16.!

3 #27$

!3

17.

3 #0.

064

$0.

418

.5 #

32$2

19.

4 #81$

320

.#y2 $

| y|

21.

3 #12

5s3

$5s

22.#

64x6

$8| x

3 |

23.

3 #!

27a

$6 $

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224

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8 n4

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4 v$

8 $2| w

| v2

27.#

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)4$

9c2

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)2 $| a

$b|

2 " 34 & 9

©G

lenc

oe/M

cGra

w-Hi

ll26

6G

lenc

oe A

lgeb

ra 2

Use

a c

alcu

lato

r to

ap

pro

xim

ate

each

val

ue

to t

hre

e d

ecim

al p

lace

s.

1.#

7.8

$2.

!#

89$3.

3 #25$

4.3 #

!4

$2.

793

!9.

434

2.92

4!

1.58

75.

4 #1.

1$

6.5 #

!0.

1$

7.6 #

5555

$8.

4 #(0

.94)

$2 $

1.02

4!

0.63

14.

208

0.97

0

Sim

pli

fy.

9.#

0.81

$10

.!#

324

$11

.!4 #

256

$12

.6 #

64$0.

9!

18!

42

13.

3 #!

64$

14.

3 #0.

512

$15

.5 #

!24

3$

16.!

4 #12

96$

!4

0.8

!3

!6

17.

5 '(18

.5 #

243x

1$

0 $19

.#(1

4a)2

$20

.#!

(14a

$)2 $

not a

!"4 3"

3x2

14| a|

real

num

ber

21.#

49m

2 t$

8 $22

. '(23

.3 #

!64

r6$

w15 $

24.#

(2x)

8$

7| m| t4

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2 w5

16x4

25.!

4 #62

5s8

$26

.3 #

216p

3$

q9 $27

.#67

6x4

$y6 $

28.

3 #!

27x9

$y1

2$

!5s

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q326

x2| y

3 |!

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4

29.!

#14

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8 n6

$30

.5 #

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x5$

y10

$31

.6 #

(m%

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6$

32.

3 #(2

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$1)

3$

!12

m4 | n

3 |!

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| m$

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33.!

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34.

4 #(x

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35.

3 #34

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$36

.#x2

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$0x

%2

$5$

!7| a

5 | b8

(x!

5)2

7d2

| x$

5|37

.RA

DIA

NT

TEM

PER

ATU

RE

The

rmal

sen

sors

mea

sure

an

obje

ct’s

rad

iant

tem

pera

ture

,w

hich

is t

he a

mou

nt o

f en

ergy

rad

iate

d by

the

obj

ect.

The

inte

rnal

tem

pera

ture

of

an

obje

ct is

cal

led

its

kine

tic

tem

pera

ture

.The

form

ula

T r"

T k4 #

e$re

late

s an

obj

ect’s

rad

iant

tem

pera

ture

Tr

to it

s ki

neti

c te

mpe

ratu

re T

k.T

he v

aria

ble

ein

the

for

mul

a is

a m

easu

reof

how

wel

l the

obj

ect

radi

ates

ene

rgy.

If a

n ob

ject

’s k

inet

ic t

empe

ratu

re is

30°

C a

nd

e"

0.94

,wha

t is

the

obj

ect’s

rad

iant

tem

pera

ture

to

the

near

est

tent

h of

a d

egre

e?29

.5)C

38.H

ERO

’S F

OR

MU

LASa

lvat

ore

is b

uyin

g fe

rtili

zer

for

his

tria

ngul

ar g

arde

n.H

e kn

ows

the

leng

ths

of a

ll th

ree

side

s,so

he

is u

sing

Her

o’s

form

ula

to f

ind

the

area

.Her

o’s

form

ula

stat

es t

hat

the

area

of

a tr

iang

le is

#s(

s!

$a)

(s!

$b)

(s!

$c)$

,whe

re a

,b,a

nd c

are

the

leng

ths

of t

he s

ides

of

the

tria

ngle

and

sis

hal

f th

e pe

rim

eter

of

the

tria

ngle

.If

the

leng

ths

of t

he s

ides

of

Salv

ator

e’s

gard

en a

re 1

5 fe

et,1

7 fe

et,a

nd 2

0 fe

et,w

hat

is t

hear

ea o

f th

e ga

rden

? R

ound

you

r an

swer

to

the

near

est

who

le n

umbe

r.12

4 ft2

4|m

|"

516m

2&

25

!10

24&

243

Pra

ctic

e (A

vera

ge)

Root

s of

Rea

l Num

bers

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-5

5-5



Rea

din

g t

o L

earn

Math

emati

csRo

ots

of R

eal N

umbe

rs

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-5

5-5

©G

lenc

oe/M

cGra

w-Hi

ll26

7G

lenc

oe A

lgeb

ra 2

Lesson 5-5

Pre-

Act

ivit

yH

ow d

o sq

uar

e ro

ots

app

ly t

o oc

ean

ogra

ph

y?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

5 at

the

top

of

page

245

in y

our

text

book

.

Supp

ose

the

leng

th o

f a

wav

e is

5 f

eet.

Exp

lain

how

you

wou

ld e

stim

ate

the

spee

d of

the

wav

e to

the

nea

rest

ten

th o

f a

knot

usi

ng a

cal

cula

tor.

(Do

not

actu

ally

cal

cula

te t

he s

peed

.)Sa

mpl

e an

swer

:Usi

ng a

cal

cula

tor,

find

the

posi

tive

squa

re ro

ot o

f 5.M

ultip

ly th

is n

umbe

r by

1.34

.Th

en ro

und

the

answ

er to

the

near

est t

enth

.

Rea

din

g t

he

Less

on

1.Fo

r ea

ch r

adic

al b

elow

,ide

ntif

y th

e ra

dica

nd a

nd t

he in

dex.

a.3 #

23$ra

dica

nd:

inde

x:

b.#

15x2

$ra

dica

nd:

inde

x:

c.5 #

!34

3$

radi

cand

:in

dex:

2.C

ompl

ete

the

follo

win

g ta

ble.

(Do

not

actu

ally

fin

d an

y of

the

indi

cate

d ro

ots.

)

Num

ber

Num

ber o

f Pos

itive

Nu

mbe

r of N

egat

ive

Num

ber o

f Pos

itive

Nu

mbe

r of N

egat

ive

Squa

re R

oots

Squa

re R

oots

Cube

Roo

ts

Cube

Roo

ts

271

11

0

!16

00

01

3.St

ate

whe

ther

eac

h of

the

fol

low

ing

is t

rue

or f

alse

.

a.A

neg

ativ

e nu

mbe

r ha

s no

rea

l fou

rth

root

s.tru

e

b.-

#12

1$

repr

esen

ts b

oth

squa

re r

oots

of

121.

true

c.W

hen

you

take

the

fif

th r

oot

of x

5 ,yo

u m

ust

take

the

abs

olut

e va

lue

of x

to id

enti

fyth

e pr

inci

pal f

ifth

roo

t.fa

lse

Hel

pin

g Y

ou

Rem

emb

er

4.W

hat

is a

n ea

sy w

ay t

o re

mem

ber

that

a n

egat

ive

num

ber

has

no r

eal s

quar

e ro

ots

but

has

one

real

cub

e ro

ot?

Sam

ple

answ

er:T

he s

quar

e of

a p

ositi

ve o

r neg

ativ

enu

mbe

r is

posi

tive,

so th

ere

is n

o re

al n

umbe

r who

se s

quar

e is

neg

ativ

e.Ho

wev

er,t

he c

ube

of a

neg

ativ

e nu

mbe

r is

nega

tive,

so a

neg

ativ

enu

mbe

r has

one

real

cub

e ro

ot,w

hich

is a

neg

ativ

e nu

mbe

r.

5!

343

215

x23

23

©G

lenc

oe/M

cGra

w-Hi

ll26

8G

lenc

oe A

lgeb

ra 2

Appr

oxim

atin

g Sq

uare

Roo

tsC

onsi

der

the

follo

win

g ex

pans

ion.

(a%& 2b a&

)2"

a2%

&2 2a ab &%

& 4b a2 2&

"a2

%b

%& 4b a2 2&

Thi

nk w

hat

happ

ens

if a

is v

ery

grea

t in

com

pari

son

to b

.The

ter

m & 4b a2 2&

is v

ery

smal

l and

can

be

disr

egar

ded

in a

n ap

prox

imat

ion.

(a%& 2b a&

)2)

a2

%b

a%

& 2b a&)

#a2

%b

$

Supp

ose

a nu

mbe

r ca

n be

exp

ress

ed a

s a2

%b,

a ,

b.T

hen

an a

ppro

xim

ate

valu

e

of t

he s

quar

e ro

ot is

a%

& 2b a&.Y

ou s

houl

d al

so s

ee t

hat

a!

& 2b a&)

#a2

!b

$.

Use

th

e fo

rmu

la !

a2

("

b"#

a(

" 2b a"to

ap

pro

xim

ate

!10

1"

and

!62

2"

.

a.#

101

$"

#10

0%

$1$

"#

102

%$

1$b.

#62

2$

"#

625

!$

3$"

#25

2!

$3$

Let

a"

10 a

nd b

"1.

Let

a"

25 a

nd b

"3.

#10

1$

) 1

0 %

& 2(1 10

)&

#62

2$

) 2

5 !

& 2(3 25

)&

) 1

0.05

) 2

4.94

Use

th

e fo

rmu

la t

o fi

nd

an

ap

pro

xim

atio

n f

or e

ach

squ

are

root

to

the

nea

rest

hu

nd

red

th.C

hec

k y

our

wor

k w

ith

a c

alcu

lato

r.

1.#

626

$25

.02

2.#

99$9.

953.

#40

2$

20.0

5

4.#

1604

$40

.05

5.#

223

$14

.93

6.#

80$8.

94

7.#

4890

$69

.93

8.#

2505

$50

.05

9.#

3575

$59

.79

10.

#1,

441

$,1

00$

1200

.42

11.

#29

0$

17.0

312

.#

260

$16

.12

13.

Show

tha

t a

!& 2b a&

) #

a2!

b$

for

a ,

b.

$a!

" 2b a"'2

&a2

!b

$" 4b a2 2"

;

disr

egar

d " 4b a2 2"

; $a!

" 2b a"'2

#a2

!b;

a!

" 2b a"#

!a2

!b

"

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-5

5-5

Exam

ple

Exam

ple


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Radi

cal E

xpre

ssio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-6

5-6

©G

lenc

oe/M

cGra

w-Hi

ll26

9G

lenc

oe A

lgeb

ra 2

Lesson 5-6

Sim

plif

y R

adic

al E

xpre

ssio

ns

For a

ny re

al n

umbe

rs a

and

b, a

nd a

ny in

tege

r n,

1:Pr

oduc

t Pro

pert

y of

Rad

ical

s1.

if n

is ev

en a

nd a

and

bar

e bo

th n

onne

gativ

e, th

en n #

ab$"

n #a$

$n #

b$.2.

if n

is od

d, th

en n #

ab$"

n #a$

$n #

b$.

To s

impl

ify

a sq

uare

roo

t,fo

llow

the

se s

teps

:1.

Fact

or t

he r

adic

and

into

as

man

y sq

uare

s as

pos

sibl

e.2.

Use

the

Pro

duct

Pro

pert

y to

isol

ate

the

perf

ect

squa

res.

3.Si

mpl

ify

each

rad

ical

.

Quo

tient

Pro

pert

y of

Rad

ical

sFo

r any

real

num

bers

aan

d b

#0,

and

any

inte

ger n

,1,

n '(

", i

f all

root

s ar

e de

fined

.

To e

limin

ate

radi

cals

fro

m a

den

omin

ator

or

frac

tion

s fr

om a

rad

ican

d,m

ulti

ply

the

num

erat

or a

nd d

enom

inat

or b

y a

quan

tity

so

that

the

rad

ican

d ha

s an

exa

ct r

oot.

n #a$

& n #b$

a & b

Sim

pli

fy 3 !

!16

a"

5 b7

".

3 #!

16a

$5 b

7$

"3 #

(!2)

3$

$2

$a

$3

$a2

$$

(b2 )

3$

$b

$"

!2a

b23 #

2a2 b

$

Sim

pli

fy %&

.

'("

Quo

tient

Pro

perty

"Fa

ctor

into

squ

ares

.

"Pr

oduc

t Pro

perty

"Si

mpl

ify.

"$

Ratio

naliz

e th

ede

nom

inat

or.

"Si

mpl

ify.

2|x|

#10

xy$

&&

15y3

#5y$

& #5y$

2|x|

#2x$

& 3y2 #

5y$

2|x|

#2x$

& 3y2 #

5y$

#(2

x)2

$$

#2x$

&&

#(3

y2 )2

$$

#5y$

#(2

x)2

$$

2x$&

&#

(3y2 )

2$

$5y

$

#8x

3$

& #45

y5$

8x3

& 45y5

8x3

" 45y5

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sim

pli

fy.

1.5 #

54$15

!6"

2.4 #

32a9

b$

20 $2a

2 | b5 |

4 !2a"

3.#

75x4

y$

7 $5x

2 y3 !

5y"

4.'(

5.'(

6.3 '(

pq3 !

5p2

""

" 10p5 q

3&40

| a3 |

b!2b"

"" 14

a6 b3

&98

6!5"

"25

36 & 125

©G

lenc

oe/M

cGra

w-Hi

ll27

0G

lenc

oe A

lgeb

ra 2

Op

erat

ion

s w

ith

Rad

ical

sW

hen

you

add

expr

essi

ons

cont

aini

ng r

adic

als,

you

can

add

only

like

ter

ms

or l

ike

rad

ical

exp

ress

ion

s.T

wo

radi

cal e

xpre

ssio

ns a

re c

alle

d li

kera

dica

l ex

pres

sion

sif

bot

h th

e in

dice

s an

d th

e ra

dica

nds

are

alik

e.

To m

ulti

ply

radi

cals

,use

the

Pro

duct

and

Quo

tien

t P

rope

rtie

s.Fo

r pr

oduc

ts o

f th

e fo

rm

(a#

b$%

c#d$)

$(e#

f$%

g#h$)

,use

the

FO

IL m

etho

d.To

rat

iona

lize

deno

min

ator

s,us

e co

nju

gate

s.N

umbe

rs o

f th

e fo

rm a

#b$

%c#

d$an

d a#

b$!

c#d$,

whe

re a

,b,c

,and

dar

era

tion

al n

umbe

rs,a

re c

alle

d co

njug

ates

.The

pro

duct

of

conj

ugat

es is

alw

ays

a ra

tion

alnu

mbe

r.

Sim

pli

fy 2

!50"

$4!

500

"!

6!12

5"

.

2#50$

%4#

500

$!

6#12

5$

"2#

52$

2$

%4#

102

$5

$!

6#52

$5

$Fa

ctor

usin

g sq

uare

s.

"2

$5

$#

2$%

4 $

10 $

#5$

!6

$5

$#

5$Si

mpl

ify s

quar

e ro

ots.

"10

#2$

%40

#5$

!30

#5$

Mul

tiply.

"10

#2$

%10

#5$

Com

bine

like

radi

cals.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Radi

cal E

xpre

ssio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-6

5-6

Exam

ple1

Exam

ple1

Sim

pli

fy (2

!3"

!4!

2")(!

3"$

2!2")

.

(2#

3$!

4#2$)

(#3$

%2#

2$)"

2#3$

$#

3$%

2#3$

$2#

2$!

4#2$

$#

3$!

4#2$

$2#

2$"

6 %

4#6$

!4#

6$!

16"

!10

Sim

pli

fy

.

"$

" " "11

!5#

5$&

& 4

6 !

5#5$

%5

&&

9 !

5

6 !

2 #5$

!3#

5$%

(#5$)

2&

&&

32!

(#5$ )

2

3 !

#5$

& 3 !

#5$

2 !

#5$

& 3 %

#5$

2 !

#5$

& 3 %

#5$

2 !

!5"

" 3 $

!5"

Exam

ple2

Exam

ple2

Exam

ple3

Exam

ple3

Exer

cises

Exer

cises

Sim

pli

fy.

1.3 #

2$%

#50$

!4#

8$2.

#20$

%#

125

$!

#45$

3.#

300

$!

#27$

!#

75$

04!

5"2!

3"

4.3 #

81$$

3 #24$

5.3 #

2$(3 #

4$%

3 #12$

)6.

2#3$(

#15$

%#

60$)

63 !

9"2

$2

3 !3"

18!

5"

7.(2

%3#

7$)(4

%#

7$)8.

(6#

3$!

4#2$)

(3#

3$%

#2$)

9.(4

#2$

!3#

5$)(2

#20

%$

5$)

29 $

14!

7"46

!6!

6"40

!2"

!30

!5"

10.

511

.5

$3!

2"12

.13

!3"

!23

"" 11

5 !

3#3$

&&

1 %

2#3$

4 %

#2$

& 2 !

#2$

5#48$

%#

75$&

&5#

3$



Skil

ls P

ract

ice

Radi

cal E

xpre

ssio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-6

5-6

©G

lenc

oe/M

cGra

w-Hi

ll27

1G

lenc

oe A

lgeb

ra 2

Lesson 5-6

Sim

pli

fy.

1.#

24$2!

6"2.

#75$

5!3"

3.3 #

16$2

3 !2"

4.!

4 #48$

!2

4 !3"

5.4#

50x5

$20

x2!

2x"6.

4 #64

a4b

$4 $

2| ab|

4 !4"

7.3 '(!

d2 f

5!

f3 !

d2 f

2"

8.'(

s2t

| s| !

t"

9.!'(

!10

.3 '(

11. '(&2 5g z3 &

12.(

3 #3$)

(5#

3$)45

13.(

4 #12$

)(3#

20$)

48!

15"14

.#2$

%#

8$%

#50$

8!2"

15.#

12$!

2#3$

%#

108

$6!

3"16

.8#

5$!

#45$

!#

80$!

5"

17.2

#48$

!#

75$!

#12$

!3"

18.(

2 %

#3$)

(6 !

#2$)

12 !

2!2"

$6!

3"!

!6"

19.(

1 !

#5$)

(1 %

#5$)

!4

20.(

3 !

#7$)

(5 %

#2$)

15 $

3!2"

!5!

7"!

!14"

21.(

#2$

!#

6$)2

8 !

4!3"

22.

23.

24.

40 $

5!6"

"" 58

5& 8

!#

6$12

!4!

2""

" 74

& 3 %

#2$

21 $

3!2"

"" 47

3& 7

!#

2$

g!10

gz"

"" 5z

3 !6 "

"3

2 & 9!

21""

73 & 7

5 " 625 & 36

1 " 21 & 8

©G

lenc

oe/M

cGra

w-Hi

ll27

2G

lenc

oe A

lgeb

ra 2

Sim

pli

fy.

1.#

540

$6!

15"2.

3 #!

432

$!

63 !

2"3.

3 #12

8$

43 !

2"

4.!

4 #40

5$

!3

4 !5"

5.3 #

!50

0$

0$!

103 !

5"6.

5 #!

121

$5$

!3

5 !5"

7.3 #

125t

6 w$

2 $5t

23 !

w2

"8.

4 #48

v8z

$13 $

2v2 z

34 !

3z"9.

3 #8g

3 k8

$2g

k23 !

k2 "

10.#

45x3

y$

8 $3x

y4!

5x"11

. '(12

.3 '(

3 !9"

13. '(

c4d

7c2

d3 !

2d"14

. '(15

.4 '(

16.(

3#15$

)(!4#

45$)

17.(

2#24$

)(7#

18$)

18.#

810

$%

#24

0$

!#

250

$!

180!

3"16

8!3"

4!10"

$4!

15"

19.6

#20$

%8#

5$!

5#45$

20.8

#48$

!6#

75$%

7#80$

21.(

3#2$

%2#

3$)2

5!5"

2!3" $

28!

5"30

$12

!6"

22.(

3 !

#7$)

223

.(#

5$!

#6$)

(#5$

%#

2$)24

.(#

2$%

#10$

)(#2$

!#

10$)

16 !

6!7"

5 $

!10"

!!

30"!

2!3"

!8

25.(

1 %

#6$)

(5 !

#7$)

26.(

#3$

%4#

7$)2

27.(

#10

8$

!6#

3$)2

5 !

!7"

$5!

6"!

!42"

115

$8!

21"0

28.

!15"

$2!

3"29

.6!

2"$

630

.

31.

32.

27 $

11!

6"33

.

34.B

RA

KIN

GT

he f

orm

ula

s"

2#5!$

esti

mat

es t

he s

peed

sin

mile

s pe

r ho

ur o

f a

car

whe

nit

leav

es s

kid

mar

ks !

feet

long

.Use

the

for

mul

a to

wri

te a

sim

plif

ied

expr

essi

on f

or s

if

!"

85.T

hen

eval

uate

sto

the

nea

rest

mile

per

hou

r.10

!17"

;41

mi/h

35.P

YTH

AG

OR

EAN

TH

EOR

EMT

he m

easu

res

of t

he le

gs o

f a

righ

t tr

iang

le c

an b

ere

pres

ente

d by

the

exp

ress

ions

6x2

yan

d 9x

2 y.U

se t

he P

ytha

gore

an T

heor

em t

o fi

nd a

sim

plif

ied

expr

essi

on f

or t

he m

easu

re o

f th

e hy

pote

nuse

.3x

2 | y| !

13"

6 $

5!x"

$x

""

4 !

x3

%#

x$& 2

!#

x$3

%#

6$&

&5

!#

24$8

$5!

2""

" 23

%#

2$& 2

!#

2$

17 !

!3"

"" 13

5 %

#3$

& 4 %

#3$

6& #

2$!

1#

3$& #

5$!

2

4 !72

a"

"3a

8& 9a

33a

2 !a"

"8b

29a

5& 64

b41 " 16

1& 12

8

216

& 24!

11""

311 & 9

Pra

ctic

e (A

vera

ge)

Radi

cal E

xpre

ssio

ns

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-6

5-6


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csRa

dica

l Exp

ress

ions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-6

5-6

©G

lenc

oe/M

cGra

w-Hi

ll27

3G

lenc

oe A

lgeb

ra 2

Lesson 5-6

Pre-

Act

ivit

yH

ow d

o ra

dic

al e

xpre

ssio

ns

app

ly t

o fa

llin

g ob

ject

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

6 at

the

top

of

page

250

in y

our

text

book

.

Des

crib

e ho

w y

ou c

ould

use

the

for

mul

a gi

ven

in y

our

text

book

and

aca

lcul

ator

to

find

the

tim

e,to

the

nea

rest

ten

th o

f a

seco

nd,t

hat

it w

ould

take

for

the

wat

er b

allo

ons

to d

rop

22 f

eet.

(Do

not

actu

ally

cal

cula

te t

heti

me.

)Sa

mpl

e an

swer

:Mul

tiply

22

by 2

(giv

ing

44) a

nd d

ivid

eby

32.

Use

the

calc

ulat

or to

find

the

squa

re ro

ot o

f the

resu

lt.Ro

und

this

squ

are

root

to th

e ne

ares

t ten

th.

Rea

din

g t

he

Less

on

1.C

ompl

ete

the

cond

itio

ns t

hat

mus

t be

met

for

a ra

dica

l exp

ress

ion

to b

e in

sim

plifi

ed fo

rm.

•T

he

nis

as

as p

ossi

ble.

•T

he

cont

ains

no

(oth

er t

han

1) t

hat

are

nth

pow

ers

of a

(n)

or p

olyn

omia

l.

•T

he r

adic

and

cont

ains

no

.

•N

o ap

pear

in t

he

.

2.a.

Wha

t ar

e co

njug

ates

of

radi

cal e

xpre

ssio

ns u

sed

for?

to ra

tiona

lize

bino

mia

lde

nom

inat

ors

b.H

ow w

ould

you

use

a c

onju

gate

to

sim

plif

y th

e ra

dica

l exp

ress

ion

?

Mul

tiply

num

erat

or a

nd d

enom

inat

or b

y 3

$!

2".c.

In o

rder

to

sim

plif

y th

e ra

dica

l exp

ress

ion

in p

art

b,tw

o m

ulti

plic

atio

ns a

re

nece

ssar

y.T

he m

ulti

plic

atio

n in

the

num

erat

or w

ould

be

done

by

the

met

hod,

and

the

mul

tipl

icat

ion

in t

he d

enom

inat

or w

ould

be

done

by

find

ing

the

of

.

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o an

othe

r pe

rson

.Whe

n ra

tion

aliz

ing

the

deno

min

ator

in t

he e

xpre

ssio

n ,m

any

stud

ents

thi

nk t

hey

shou

ld m

ulti

ply

num

erat

or

and

deno

min

ator

by

.How

wou

ld y

ou e

xpla

in t

o a

clas

smat

e w

hy t

his

is in

corr

ect

and

wha

t he

sho

uld

do in

stea

d.Sa

mpl

e an

swer

:Bec

ause

you

are

wor

king

with

cube

root

s,no

t squ

are

root

s,yo

u ne

ed to

mak

e th

e ra

dica

nd in

the

deno

min

ator

a p

erfe

ct c

ube,

not a

per

fect

squ

are.

Mul

tiply

num

erat

or a

nd

deno

min

ator

by

to

mak

e th

e de

nom

inat

or 3 !

8",w

hich

equ

als

2.3 !

4"" 3 !

4 "

3 #2$

& 3 #2$

1& 3 #

2$

squa

res

two

diffe

renc

e

FOIL

1 %

#2$

& 3 !

#2$

deno

min

ator

radi

cals

fract

ions

inte

ger

fact

ors

radi

cand

smal

lin

dex

©G

lenc

oe/M

cGra

w-Hi

ll27

4G

lenc

oe A

lgeb

ra 2

Spec

ial P

rodu

cts

with

Rad

ical

sN

otic

e th

at (#

3$)(#

3$)"

3,or

(#3$)

2"

3.

In g

ener

al,(

#x$)

2"

xw

hen

x .

0.

Als

o,no

tice

tha

t (#

9$)(#

4$)"

#36$

.

In g

ener

al,(

#x$)

(#y$)

"#

xy$w

hen

xan

d y

are

not

nega

tive

.

You

can

use

thes

e id

eas

to f

ind

the

spec

ial p

rodu

cts

belo

w.

(#a$

%#

b$)(#

a$!

#b$)

"(#

a$)2

!(#

b$)2

"a

!b

(#a$

%#

b$)2

"(#

a$)2

%2#

ab$%

(#b$)

2"

a%

2#ab$

%b

(#a$

!#

b$)2

"(#

a$)2

!2#

ab$%

(#b$)

2"

a!

2#ab$

%b

Fin

d t

he

pro

du

ct:(

!2"

$!

5")( !

2"!

!5")

.

(#2$

%#

5$)(#

2$!

#5$)

"(#

2$)2

!(#

5$)2

"2

!5

"!

3

Eva

luat

e ( !

2"$

!8")

2 .

(#2$

%#

8$)2

"(#

2$)2

%2#

2$#8$

%(#

8$)2

"2

%2#

16$%

8 "

2 %

2(4)

%8

"2

%8

%8

"18

Mu

ltip

ly.

1.(#

3$!

#7$)

(#3$

%#

7$)!

42.

(#10$

%#

2$)(#

10$!

#2$)

8

3.(#

2x$!

#6$)

(#2x$

!#

6$)2x

!6

4.(#

3$!

27)2

12

5.(#

1000

$%

#10$

)212

106.

(#y$

%#

5$)(#

y$!

#5$)

y!

5

7.(#

50$!

#x$)

250

!10

!2x"

$x

8.(#

x$%

20)2

x$

4!5x"

$20

You

can

exte

nd t

hese

idea

s to

pat

tern

s fo

r su

ms

and

diff

eren

ces

of c

ubes

.St

udy

the

patt

ern

belo

w.

(3 #8$

! 3 #

x$)(3 #

82 $%

3 #8x$

%3 #

x2 $)"

3 #83 $

! 3 #

x3 $"

8 !

x

Mu

ltip

ly.

9.(3 #

2$!

3 #5$)

(3 #22 $

%3 #

10$%

3 #52 $

)!

3

10.(

3 #y$

%3 #

w$)(

3 #y2 $

!3 #

yw$%

3 #w

2 $)

y$

w

11.

(3 #7$

%3 #

20$)(

3 #72 $

!3 #

140

$%

3 #20

2$

)27

12.(

3 #11$

!3 #

8$)(3 #

112

$%

3 #88$

%3 #

82 $)

3

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-6

5-6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2



Stu

dy

Gu

ide

and I

nte

rven

tion

Ratio

nal E

xpon

ents

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-7

5-7

©G

lenc

oe/M

cGra

w-Hi

ll27

5G

lenc

oe A

lgeb

ra 2

Lesson 5-7

Rat

ion

al E

xpo

nen

ts a

nd

Rad

ical

s

Defin

ition

of b

" n1 "Fo

r any

real

num

ber b

and

any

posit

ive in

tege

r n,

b&1 n&"

n #b$,

exc

ept w

hen

b)

0 an

d n

is ev

en.

Defin

ition

of b

"m n"Fo

r any

non

zero

real

num

ber b

, and

any

inte

gers

man

d n,

with

n,

1,

b&m n&"

n #bm $

"(n #

b$)m, e

xcep

t whe

n b

)0

and

nis

even

.

Wri

te 2

8"1 2"in

rad

ical

for

m.

Not

ice

that

28

,0.

28&1 2&

"#

28$"

#22

$7

$"

#22 $

$#

7$"

2#7$

Eva

luat

e $

'"1 3" .

Not

ice

that

!8

)0,

!12

5 )

0,an

d 3

is o

dd.

!"&1 3&

" " "2 & 5!

2& !

53 #!

8$

& 3 #!

125

$!

8& !

125

!8

" !12

5Ex

ampl

e1Ex

ampl

e1Ex

ampl

e2Ex

ampl

e2

Exer

cises

Exer

cises

Wri

te e

ach

exp

ress

ion

in

rad

ical

for

m.

1.11

&1 7&2.

15&1 3&

3.30

0&3 2&

7 !11"

3 !15"

!30

03"

Wri

te e

ach

rad

ical

usi

ng

rati

onal

exp

onen

ts.

4.#

47$5.

3 #3a

5 b2

$6.

4 #16

2p5

$

47"1 2"

3"1 3" a"5 3" b"2 3"3

*2"1 4"

*p"5 4"

Eva

luat

e ea

ch e

xpre

ssio

n.

7.!

27&2 3&

8.9.

(0.0

004)

&1 2&

90.

02

10.8

&2 3&$

4&3 2&11

.12

.

321 " 2

1 " 4

16!

&1 2&

& (0.2

5)&1 2&

144!

&1 2&

& 27!

&1 3&

1 " 105!&1 2&

& 2#5$

©G

lenc

oe/M

cGra

w-Hi

ll27

6G

lenc

oe A

lgeb

ra 2

Sim

plif

y Ex

pre

ssio

ns

All

the

prop

erti

es o

f po

wer

s fr

om L

esso

n 5-

1 ap

ply

to r

atio

nal

expo

nent

s.W

hen

you

sim

plif

y ex

pres

sion

s w

ith

rati

onal

exp

onen

ts,l

eave

the

exp

onen

t in

rati

onal

for

m,a

nd w

rite

the

exp

ress

ion

wit

h al

l pos

itiv

e ex

pone

nts.

Any

exp

onen

ts in

the

deno

min

ator

mus

t be

pos

itiv

e in

tege

rs

Whe

n yo

u si

mpl

ify

radi

cal e

xpre

ssio

ns,y

ou m

ay u

se r

atio

nal e

xpon

ents

to

sim

plif

y,bu

t yo

uran

swer

sho

uld

be in

rad

ical

for

m.U

se t

he s

mal

lest

inde

x po

ssib

le.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Ratio

nal E

xpon

ents

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-7

5-7

Sim

pli

fy y

"2 3"*

y"3 8" .

y&2 3&$

y&3 8&"

y&2 3&%

&3 8&"

y&2 25 4&

Sim

pli

fy 4 !

144x

6"

.4 #

144x

6$

"(1

44x6

)&1 4&

"(2

4$

32$

x6)&1 4&

"(2

4 )&1 4&

$(3

2 )&1 4&

$(x

6 )&1 4&

"2

$3&1 2&

$x&3 2&

"2x

$(3

x)&1 2&

"2x

#3x$

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sim

pli

fy e

ach

exp

ress

ion

.

1.x&4 5&

$x&6 5&

2.! y&2 3& "&3 4&

3.p&4 5&

$p& 17 0&

x2y"1 2"

p"3 2"

4.! m

!&6 5& "&2 5&

5.x!

&3 8&$

x&4 3&6.

! s!&1 6& "!

&4 3&

x"2 23 4"s"2 9"

7.8.

! a&2 3& "&6 5&$! a&2 5& "3

9.

p"2 3"a2

10.

6 #12

8$

11.

4 #49$

12.

5 #28

8$

2!2"

!7"

25 !

9"

13.#

32$$

3#16$

14.

3 #25$

$#

125

$15

.6 #

16$

48!

2"25

6 !5"

3 !4"

16.

17.#

$3 #48$

18.

6 !48"

!a"

6 !b5 "

"b

x !3"

!6 !

35 ""

" 6

a3 #

b4 $& #

ab3

$x

!3 #

3$&

#12$

x"5 6"

" xx!&1 2&

& x!&1 3&

p & p&1 3&1 " m3


An

swer

s


Skil

ls P

ract

ice

Ratio

nal E

xpon

ents

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-7

5-7

©G

lenc

oe/M

cGra

w-Hi

ll27

7G

lenc

oe A

lgeb

ra 2

Lesson 5-7

Wri

te e

ach

exp

ress

ion

in

rad

ical

for

m.

1.3&1 6&

6 !3"

2.8&1 5&

5 !8"

3.12

&2 3&3 !

122

"or

(3 !12"

)24.

(s3)&3 5&

s5 !

s4 "

Wri

te e

ach

rad

ical

usi

ng

rati

onal

exp

onen

ts.

5.#

51$51

"1 2"6.

3 #37$

37"1 3"

7.4 #

153

$15

"3 4"8.

3 #6x

y2$

6"1 3" x"1 3" y"2 3"

Eva

luat

e ea

ch e

xpre

ssio

n.

9.32

&1 5&2

10.8

1&1 4&3

11.2

7!&1 3&

12.4

!&1 2&

13.1

6&3 2&64

14.(

!24

3)&4 5&

81

15.2

7&1 3&$

27&5 3&

729

16. !

"&3 2&

Sim

pli

fy e

ach

exp

ress

ion

.

17.c

&1 52 &$

c&3 5&c3

18.m

&2 9&$

m&1 96 &

m2

19.! q

&1 2& "3q"3 2"

20.p

!&1 5&

21.x

!& 16 1&

22.

x" 15 2"

23.

24.

25.12 #

64$!

2"26

.8 #

49a8

b$

2 $| a

|4 !7b"

n"2 3"

" nn&1 3&

& n&1 6&$

n&1 2&

y"1 4"

" yy!

&1 2&

& y&1 4&

x&2 3&

& x&1 4&

x" 15 1"

" x

p"4 5"

" p8 " 274 & 9

1 " 21 " 3

©G

lenc

oe/M

cGra

w-Hi

ll27

8G

lenc

oe A

lgeb

ra 2

Wri

te e

ach

exp

ress

ion

in

rad

ical

for

m.

1.5&1 3&

2.6&2 5&

3.m

&4 7&4.

(n3 )&2 5&

3 !5"

5 !62 "

or (5 !

6")27 !

m4

"or

(7 !m"

)4n

5 !n"

Wri

te e

ach

rad

ical

usi

ng

rati

onal

exp

onen

ts.

5.#

79$6.

4 #15

3$

7.3 #

27m

6 n$

4 $8.

5#2a

10b

$

79"1 2"

153"1 4"

3m2 n

"4 3"5

*2"1 2" | a

5 | b"1 2"

Eva

luat

e ea

ch e

xpre

ssio

n.

9.81

&1 4&3

10.1

024!

&1 5&11

.8!

&5 3&

12.!

256!

&3 4&!

13.(

!64

)!&2 3&

14.2

7&1 3&$

27&4 3&

243

15. !

"&2 3&16

.17

.! 25&1 2& "! !

64!

&1 3& " !

Sim

pli

fy e

ach

exp

ress

ion

.

18.g

&4 7&$

g&3 7&g

19.s

&3 4&$

s&1 43 &s4

20.! u

!&1 3& "!

&4 5&u" 14 5"

21.y

!&1 2&

22.b

!&3 5&

23.

q"1 5"24

.25

.

26.10 #

85 $2!

2"27

.#12$

$5 #

123

$28

.4 #

6$$

34 #

6$29

.

1210 !

12"3!

6"

30.E

LEC

TRIC

ITY

The

am

ount

of

curr

ent

in a

mpe

res

Ith

at a

n ap

plia

nce

uses

can

be

calc

ulat

ed u

sing

the

for

mul

a I

"!

"&1 2& ,whe

re P

is t

he p

ower

in w

atts

and

Ris

the

resi

stan

ce in

ohm

s.H

ow m

uch

curr

ent

does

an

appl

ianc

e us

e if

P"

500

wat

ts a

nd

R"

10 o

hms?

Rou

nd y

our

answ

er t

o th

e ne

ares

t te

nth.

7.1

amps

31.B

USI

NES

SA

com

pany

tha

t pr

oduc

es D

VD

s us

es t

he f

orm

ula

C"

88n&1 3&

%33

0 to

calc

ulat

e th

e co

st C

in d

olla

rs o

f pr

oduc

ing

nD

VD

s pe

r da

y.W

hat

is t

he c

ompa

ny’s

cos

tto

pro

duce

150

DV

Ds

per

day?

Rou

nd y

our

answ

er t

o th

e ne

ares

t do

llar.

$798

P & R

a!3b"

"3b

a& #

3b$

2z$

2z"1 2"

""

z!

12z

&1 2&

& z&1 2&!

1

t"1 11 2"

" 5t&2 3&

& 5t&1 2&

$t!

&3 4&

q&3 5&

& q&2 5&

b"2 5"

" b

y"1 2"

" y

5 " 416 " 49

64&2 3&

& 343&2 3&

25 " 3612

5& 21

6

1 " 161 " 64

1 " 321 " 4

Pra

ctic

e (A

vera

ge)

Ratio

nal E

xpon

ents

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-7

5-7



Rea

din

g t

o L

earn

Math

emati

csRa

tiona

l Exp

onen

ts

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-7

5-7

©G

lenc

oe/M

cGra

w-Hi

ll27

9G

lenc

oe A

lgeb

ra 2

Lesson 5-7

Pre-

Act

ivit

yH

ow d

o ra

tion

al e

xpon

ents

ap

ply

to

astr

onom

y?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

7 at

the

top

of

page

257

in y

our

text

book

.

The

form

ula

in t

he in

trod

ucti

on c

onta

ins

the

expo

nent

.W

hat

do y

ou t

hink

it

mig

ht m

ean

to r

aise

a n

umbe

r to

the

po

wer

?

Sam

ple

answ

er:T

ake

the

fifth

root

of t

he n

umbe

r and

squ

are

it.

Rea

din

g t

he

Less

on

1.C

ompl

ete

the

follo

win

g de

fini

tion

s of

rat

iona

l exp

onen

ts.

•Fo

r an

y re

al n

umbe

r b

and

for

any

posi

tive

inte

ger

n,b& n1 &

"ex

cept

whe

n b

and

nis

.

•Fo

r an

y no

nzer

o re

al n

umbe

r b,

and

any

inte

gers

man

d n,

wit

h n

,

b&m n&"

"

,exc

ept

whe

n b

and

nis

.

2.C

ompl

ete

the

cond

itio

ns t

hat

mus

t be

met

in o

rder

for

an

expr

essi

on w

ith

rati

onal

expo

nent

s to

be

sim

plif

ied.

•It

has

no

expo

nent

s.

•It

has

no

expo

nent

s in

the

.

•It

is n

ot a

fr

acti

on.

•T

he

of a

ny r

emai

ning

is

the

nu

mbe

r po

ssib

le.

3.M

arga

rita

and

Pie

rre

wer

e w

orki

ng t

oget

her

on t

heir

alg

ebra

hom

ewor

k.O

ne e

xerc

ise

aske

d th

em t

o ev

alua

te t

he e

xpre

ssio

n 27

&4 3& .Mar

gari

ta t

houg

ht t

hat

they

sho

uld

rais

e 27

to

the

four

th p

ower

fir

st a

nd t

hen

take

the

cub

e ro

ot o

f th

e re

sult

.Pie

rre

thou

ght

that

they

sho

uld

take

the

cub

e ro

ot o

f 27

fir

st a

nd t

hen

rais

e th

e re

sult

to

the

four

th p

ower

.W

hose

met

hod

is c

orre

ct?

Both

met

hods

are

cor

rect

.

Hel

pin

g Y

ou

Rem

emb

er

4.So

me

stud

ents

hav

e tr

oubl

e re

mem

beri

ng w

hich

par

t of

the

fra

ctio

n in

a r

atio

nal

expo

nent

giv

es t

he p

ower

and

whi

ch p

art

give

s th

e ro

ot.H

ow c

an y

our

know

ledg

e of

inte

ger

expo

nent

s he

lp y

ou t

o ke

ep t

his

stra

ight

?Sa

mpl

e an

swer

:An

inte

ger

expo

nent

can

be

writ

ten

as a

ratio

nal e

xpon

ent.

For e

xam

ple,

23&

2"3 1" .Yo

u kn

ow th

at th

is m

eans

that

2 is

rais

ed to

the

third

pow

er,s

o th

enu

mer

ator

mus

t giv

e th

e po

wer

,and

,the

refo

re,t

he d

enom

inat

or m

ust

give

the

root

.

leas

tra

dica

lin

dexco

mpl

exde

nom

inat

orfra

ctio

nal

nega

tive

even

+0

(n !b")

mn !

bm ",

1ev

en+

0

n !b "

2 & 5

2 & 5

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lenc

oe/M

cGra

w-Hi

ll28

0G

lenc

oe A

lgeb

ra 2

Less

er-K

now

n G

eom

etric

For

mul

asM

any

geom

etri

c fo

rmul

as in

volv

e ra

dica

l exp

ress

ions

.

Mak

e a

dra

win

g to

ill

ust

rate

eac

h o

f th

e fo

rmu

las

give

n o

n t

his

pag

e.T

hen

eva

luat

e th

e fo

rmu

la f

or t

he

give

n v

alu

e of

th

e va

riab

le.R

oun

d

answ

ers

to t

he

nea

rest

hu

nd

red

th.

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-7

5-7

1.T

he a

rea

of a

n is

osce

les

tria

ngle

.Tw

osi

des

have

leng

th a

;the

oth

er s

ide

has

leng

th c

.Fin

d A

whe

n a

"6

and

c"

7.

A"

& 4c & #4a

2!

$c2 $

A(

17.0

6

3.T

he a

rea

of a

reg

ular

pen

tago

n w

ith

asi

de o

f le

ngth

a.F

ind

Aw

hen

a"

4.

A"

&a 42 &#

25 %

$10

#5$

$A

(27

.53

5.T

he v

olum

e of

a r

egul

ar t

etra

hedr

onw

ith

an e

dge

of le

ngth

a.F

ind

Vw

hen

a"

2.

V"

& 1a 23 &#

2$V

(0.

94

7.H

eron

’s F

orm

ula

for

the

area

of

a tr

iang

le u

ses

the

sem

i-pe

rim

eter

s,

whe

re s

"&a

%2b

%c

&.T

he s

ides

of

the

tria

ngle

hav

e le

ngth

s a,

b,an

d c.

Fin

d A

whe

n a

"3,

b"

4,an

d c

"5.

A"

#s(

s!

$a)

(s!

$b)

(s!

$c)$

A&

6

2.T

he a

rea

of a

n eq

uila

tera

l tri

angl

e w

ith

a si

de o

f le

ngth

a.F

ind

Aw

hen

a"

8.

A"

&a 42 &#

3$A

(27

.71

4.T

he a

rea

of a

reg

ular

hex

agon

wit

h a

side

of

leng

th a

.Fin

d A

whe

n a

"9.

A"

&3 2a2 &#

3$A

(21

0.44

6.T

he a

rea

of t

he c

urve

d su

rfac

e of

a r

ight

cone

wit

h an

alt

itud

e of

han

d ra

dius

of

base

r.F

ind

S w

hen

r"

3 an

d h

"6.

S"

+r#

r2%

h$

2 $S

(63

.22

8.T

he r

adiu

s of

a c

ircl

e in

scri

bed

in a

giv

entr

iang

le a

lso

uses

the

sem

i-pe

rim

eter

.F

ind

rw

hen

a"

6,b

"7,

and

c"

9.

r" r(

1.91

rb ca

#s(

s!

$a)

(s!

$b)

(s!

$c)$

&&

&s

r

h

aa

aa

aa

a

a

a

a

b

c

aa

a

aa

a

aa

a

a

a

a

c

a


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Radi

cal E

quat

ions

and

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

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OD

____

_

5-8

5-8

©G

lenc

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cGra

w-Hi

ll28

1G

lenc

oe A

lgeb

ra 2

Lesson 5-8

Solv

e R

adic

al E

qu

atio

ns

The

fol

low

ing

step

s ar

e us

ed in

sol

ving

equ

atio

ns t

hat

have

vari

able

s in

the

rad

ican

d.So

me

alge

brai

c pr

oced

ures

may

be

need

ed b

efor

e yo

u us

e th

ese

step

s.

Step

1Is

olat

e th

e ra

dica

l on

one

side

of th

e eq

uatio

n.St

ep 2

To e

limin

ate

the

radi

cal,

raise

eac

h sid

e of

the

equa

tion

to a

pow

er e

qual

to th

e in

dex

of th

e ra

dica

l.St

ep 3

Solve

the

resu

lting

equa

tion.

Step

4Ch

eck

your

sol

utio

n in

the

orig

inal

equ

atio

n to

mak

e su

re th

at y

ou h

ave

not o

btai

ned

any

extra

neou

s ro

ots.

Sol

ve 2

!4x

$"

8"!

4 &

8.

2#4x

%8

$!

4 "

8O

rigin

al e

quat

ion

2#4x

%8

$"

12Ad

d 4

to e

ach

side.

#4x

%8

$"

6Is

olat

e th

e ra

dica

l.

4x%

8 "

36Sq

uare

eac

h sid

e.

4x"

28Su

btra

ct 8

from

eac

h sid

e.

x"

7Di

vide

each

sid

e by

4.

Ch

eck

2#4(

7) %

$8$

!4

!8

2#36$

!4

!8

2(6)

!4

!8

8 "

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he s

olut

ion

x"

7 ch

ecks

.

Sol

ve !

3x$

"1"

&!

5x"!

1.

#3x

%1

$"

#5x$

!1

Orig

inal

equ

atio

n

3x%

1"

5x!

2#5x$

%1

Squa

re e

ach

side.

2#5x$

"2x

Sim

plify

.

#5x$

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ate

the

radi

cal.

5x"

x2Sq

uare

eac

h sid

e.

x2!

5x"

0Su

btra

ct 5

xfro

m e

ach

side.

x(x

!5)

"0

Fact

or.

x"

0 or

x"

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hec

k#

3(0)

%$

1$"

1,bu

t #

5(0)

$!

1 "

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so 0

isno

t a

solu

tion

.#

3(5)

%$

1$"

4,an

d #

5(5)

$!

1 "

4,so

the

solu

tion

is x

"5.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

equ

atio

n.

1.3

%2x

#3$

"5

2.2#

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4$

%1

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3.8

%#

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"2

15no

sol

utio

n

4.#

5 !

x$

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no s

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12

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08.

10 !

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9.#

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x$"

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$

512

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sol

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n

10.4

3 #2x

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$1$

!2

"10

11.2

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$%

#3x

!6

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.#9x

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814

3,4

!3"

"3

©G

lenc

oe/M

cGra

w-Hi

ll28

2G

lenc

oe A

lgeb

ra 2

Solv

e R

adic

al In

equ

alit

ies

A r

adic

al i

neq

ual

ity

is a

n in

equa

lity

that

has

a v

aria

ble

in a

rad

ican

d.U

se t

he f

ollo

win

g st

eps

to s

olve

rad

ical

ineq

ualit

ies.

Step

1If

the

inde

x of

the

root

is e

ven,

iden

tify

the

valu

es o

f the

var

iabl

e fo

r whi

ch th

e ra

dica

nd is

non

nega

tive.

Step

2So

lve th

e in

equa

lity a

lgeb

raica

lly.

Step

3Te

st v

alue

s to

che

ck y

our s

olut

ion.

Sol

ve 5

!!

20x

$"

4"-

!3.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Radi

cal E

quat

ions

and

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-8

5-8

Exam

ple

Exam

ple

Sinc

e th

e ra

dica

nd o

f a

squa

re r

oot

mus

t be

gre

ater

tha

n or

equ

al t

oze

ro,f

irst

sol

ve20

x%

4 .

0.20

x%

4 .

020

x.

!4

x.

!1 & 5

Now

sol

ve 5

!#

20x

%$

4$.

!3.

5 !

#20

x%

$4$

.!

3O

rigin

al in

equa

lity

#20

x%

$4$

(8

Isol

ate

the

radi

cal.

20x

%4

(64

Elim

inat

e th

e ra

dica

l by

squa

ring

each

sid

e.

20x

(60

Subt

ract

4 fr

om e

ach

side.

x(

3Di

vide

each

sid

e by

20.

It a

ppea

rs t

hat

!(

x(

3 is

the

sol

utio

n.Te

st s

ome

valu

es.

x&

!1

x&

0x

&4

#20

(!1

$) %

4$

is no

t a re

al

5 !

#20

(0)

$%

4$

"3,

so

the

5 !

#20

(4)

$%

4$

&!

4.2,

so

num

ber,

so th

e in

equa

lity is

ineq

uality

is s

atisf

ied.

th

e in

equa

lity is

not

not s

atisf

ied.

satis

fied

The

refo

re t

he s

olut

ion

!(

x(

3 ch

ecks

.

Sol

ve e

ach

in

equ

alit

y.

1.#

c!

2$

%4

.7

2.3#

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1$

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3.#

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9$!

2 ,

5

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x+

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3 #x

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25.

8 !

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$.

36.

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$!

4 ,

2

x+

6!

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x,

14

7.9

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49.

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5

!.

x.

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4 .

1211

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1$%

#d$

(5

12.4

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$.

10

x-

260

.d

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b-

6

6 " 51 " 2

20&

&#

3x%

1$4 " 3

1 " 2

1 & 5

1 & 5

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Radi

cal E

quat

ions

and

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-8

5-8

©G

lenc

oe/M

cGra

w-Hi

ll28

3G

lenc

oe A

lgeb

ra 2

Lesson 5-8

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.

1.#

x$"

525

2.#

x$%

3 "

716

3.5#

j$"1

4.v&1 2&

%1

"0

no s

olut

ion

5.18

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sol

utio

n6.

3 #2w$

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32

7.#

b!

5$

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218.

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1$"

58

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1110

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63

11.#

k!

4$

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4012

.(2

d%

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t!

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1114

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(1 !

7u)&1 3&

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2$

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z!

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no s

olut

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8

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no s

olut

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18.5

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s!

3$

(6

3 .

s.

4

19.!

2 %

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$)

7!

1 +

x+

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.!

6!

2 .

a.

16

21.2

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$,

10r,

722

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$,

3!

+x

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y%

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y-

3224

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!15

!.

r.2

3 " 11

1 " 3

5 " 2

1 " 25

©G

lenc

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cGra

w-Hi

ll28

4G

lenc

oe A

lgeb

ra 2

Sol

ve e

ach

equ

atio

n o

r in

equ

alit

y.

1.#

x$"

864

2.4

!#

x$"

31

3.#

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104.

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96.

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no s

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341

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8

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no s

olut

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11.#

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sol

utio

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a-

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c-

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)!

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7

29.S

TATI

STIC

SSt

atis

tici

ans

use

the

form

ula

!"

#v$

to c

alcu

late

a s

tand

ard

devi

atio

n !

,w

here

vis

the

var

ianc

e of

a d

ata

set.

Fin

d th

e va

rian

ce w

hen

the

stan

dard

dev

iati

on

is 1

5.22

5

30.G

RA

VIT

ATI

ON

Hel

ena

drop

s a

ball

from

25

feet

abo

ve a

lake

.The

for

mul

a

t"

#25

!$

h$de

scri

bes

the

tim

e t

in s

econ

ds t

hat

the

ball

is h

feet

abo

ve t

he w

ater

.

How

man

y fe

et a

bove

the

wat

er w

ill t

he b

all b

e af

ter

1 se

cond

?9

ft

1 & 4

3 " 2

7 " 2

41 " 2

3 " 47 " 4

63 " 4

1 " 1249 " 2

Pra

ctic

e (A

vera

ge)

Radi

cal E

quat

ions

and

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-8

5-8


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csRa

dica

l Equ

atio

ns a

nd In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-8

5-8

©G

lenc

oe/M

cGra

w-Hi

ll28

5G

lenc

oe A

lgeb

ra 2

Lesson 5-8

Pre-

Act

ivit

yH

ow d

o ra

dic

al e

quat

ion

s ap

ply

to

man

ufa

ctu

rin

g?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

8 at

the

top

of

page

263

in y

our

text

book

.

Exp

lain

how

you

wou

ld u

se t

he f

orm

ula

in y

our

text

book

to

find

the

cos

t of

prod

ucin

g 12

5,00

0 co

mpu

ter

chip

s.(D

escr

ibe

the

step

s of

the

cal

cula

tion

in t

heor

der

in w

hich

you

wou

ld p

erfo

rm t

hem

,but

do

not

actu

ally

do

the

calc

ulat

ion.

)

Sam

ple

answ

er:R

aise

125

,000

to th

e po

wer

by

taki

ng th

e cu

be ro

ot o

f 125

,000

and

squ

arin

g th

e re

sult

(or r

aise

125

,000

to

the

pow

er b

y en

terin

g 12

5,00

0 ^

(2/3

) on

a ca

lcul

ator

).M

ultip

ly th

e nu

mbe

r you

get

by

10 a

nd th

en a

dd 1

500.

Rea

din

g t

he

Less

on

1.a.

Wha

t is

an

extr

aneo

us s

olut

ion

of a

rad

ical

equ

atio

n?Sa

mpl

e an

swer

:a n

umbe

rth

at s

atis

fies

an e

quat

ion

obta

ined

by

rais

ing

both

sid

es o

f the

orig

inal

equa

tion

to a

hig

her p

ower

but

doe

s no

t sat

isfy

the

orig

inal

equ

atio

n

b.D

escr

ibe

two

way

s yo

u ca

n ch

eck

the

prop

osed

sol

utio

ns o

f a

radi

cal e

quat

ion

in o

rder

to d

eter

min

e w

heth

er a

ny o

f th

em a

re e

xtra

neou

s so

luti

ons.

Sam

ple

answ

er:O

new

ay is

to c

heck

eac

h pr

opos

ed s

olut

ion

by s

ubst

itutin

g it

into

the

orig

inal

equ

atio

n.An

othe

r way

is to

use

a g

raph

ing

calc

ulat

or to

gra

phbo

th s

ides

of t

he o

rigin

al e

quat

ion.

See

whe

re th

e gr

aphs

inte

rsec

t.Th

is c

an h

elp

you

iden

tify

solu

tions

that

may

be

extra

neou

s.

2.C

ompl

ete

the

step

s th

at s

houl

d be

fol

low

ed in

ord

er t

o so

lve

a ra

dica

l ine

qual

ity.

Step

1If

the

of

the

roo

t is

,i

dent

ify

the

valu

es o

f

the

vari

able

for

whi

ch t

he

is

.

Step

2So

lve

the

alge

brai

cally

.

Step

3Te

st

to c

heck

you

r so

luti

on.

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o an

othe

r pe

rson

.Sup

pose

tha

t yo

urfr

iend

Leo

ra t

hink

s th

at s

he d

oes

not

need

to

chec

k he

r so

luti

ons

to r

adic

al e

quat

ions

by

subs

titu

tion

bec

ause

she

kno

ws

she

is v

ery

care

ful a

nd s

eldo

m m

akes

mis

take

s in

her

wor

k.H

ow c

an y

ou e

xpla

in t

o he

r th

at s

he s

houl

d ne

vert

hele

ss c

heck

eve

ry p

ropo

sed

solu

tion

in t

he o

rigi

nal e

quat

ion?

Sam

ple

answ

er:S

quar

ing

both

sid

es o

f an

equa

tion

can

prod

uce

an e

quat

ion

that

is n

ot e

quiv

alen

t to

the

orig

inal

one.

For e

xam

ple,

the

only

sol

utio

n of

x&

5 is

5,b

ut th

e sq

uare

deq

uatio

n x2

&25

has

two

solu

tions

,5 a

nd !

5.

valu

esin

equa

lity

nonn

egat

ive

radi

cand

even

inde

x

2 " 3

2 " 3

©G

lenc

oe/M

cGra

w-Hi

ll28

6G

lenc

oe A

lgeb

ra 2

Trut

h Ta

bles

In m

athe

mat

ics,

the

basi

c op

erat

ions

are

add

itio

n,su

btra

ctio

n,m

ulti

plic

atio

n,di

visi

on,f

indi

ng a

roo

t,an

d ra

isin

g to

a p

ower

.In

logi

c,th

e ba

sic

oper

atio

ns

are

the

follo

win

g:no

t(*

),an

d (+

),or

(,

),an

dim

plie

s(→

).

If P

and

Qar

e st

atem

ents

,the

n *

Pm

eans

not

P;*

Qm

eans

not

Q;P

+ Q

mea

ns P

and

Q;P

,Q

mea

ns P

or Q

;and

P →

Qm

eans

Pim

plie

s Q

.The

op

erat

ions

are

def

ined

by

trut

h ta

bles

.On

the

left

bel

ow is

the

tru

th t

able

for

th

e st

atem

ent

*P

.Not

ice

that

the

re a

re t

wo

poss

ible

con

diti

ons

for

P,t

rue

(T)

or f

alse

(F).

If P

is t

rue,

*P

is f

alse

;if P

is f

alse

,*P

is t

rue.

Als

o sh

own

are

the

trut

h ta

bles

for

P +

Q,P

,Q

,and

P →

Q.

P*

PP

QP

+ Q

PQ

P ,

QP

QP

→ Q

TF

TT

TT

TT

TT

TF

TT

FF

TF

TT

FF

FT

FF

TT

FT

TF

FF

FF

FF

FT

You

can

use

this

info

rmat

ion

to f

ind

out

unde

r w

hat

cond

itio

ns a

com

plex

st

atem

ent

is t

rue. U

nd

er w

hat

con

dit

ion

s is

)P

*Q

tru

e?

Cre

ate

the

trut

h ta

ble

for

the

stat

emen

t.U

se t

he in

form

atio

n fr

om t

he t

ruth

ta

ble

abov

e fo

r P

*Q

to c

ompl

ete

the

last

col

umn.

PQ

*P

*P

,Q

TT

FT

Whe

n on

e st

atem

ent i

s T

FF

Ftru

e an

d on

e is

fals

e,F

TT

Tth

e co

njun

ctio

n is

true

.F

FT

T

The

tru

th t

able

indi

cate

s th

at *

P ,

Qis

tru

e in

all

case

s ex

cept

whe

re P

is t

rue

and

Qis

fals

e.

Use

tru

th t

able

s to

det

erm

ine

the

con

dit

ion

s u

nd

er w

hic

h e

ach

sta

tem

ent

is t

rue.

1.*

P ,

*Q

2.*

P →

(P →

Q)

all e

xcep

t whe

re b

oth

all

Pan

d Q

are

true

3.(P

,Q

),(*

P +

*Q

)4.

(P →

Q),

(Q →

P)

all

all

5.(P

→ Q

)+ (Q

→ P

)6.

(*P

+ *

Q)→

*(P

,Q

)bo

th P

and

Qar

e tru

e;al

lbo

th P

and

Qar

e fa

lse

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-8

5-8

Exam

ple

Exam

ple



To s

olve

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atio

n th

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rea

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utio

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act

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DATE

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OD

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5-9

5-9

©G

lenc

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cGra

w-Hi

ll28

7G

lenc

oe A

lgeb

ra 2

Lesson 5-9

Ad

d a

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btr

act

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mp

lex

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mb

ers

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ex n

umbe

r is

any

num

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an b

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ex N

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are

real

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iis

the

imag

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d b

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the

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tion

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Com

bine

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s.Su

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%(b

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mpl

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%(b

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ampl

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mb

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the

defin

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.

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rven

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onti

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)

Com

plex

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bers

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-9

5-9

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

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Exer

cises


An

swer

s


Skil

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ract

ice

Com

plex

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bers

NAM

E__

____

____

____

____

____

____

____

____

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____

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OD

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5-9

5-9

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pli

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LEC

TRIC

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The

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36.E

LEC

TRIC

ITY

Usi

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NAM

E__

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5-9

5-9



Rea

din

g t

o L

earn

Math

emati

csCo

mpl

ex N

umbe

rs

NAM

E__

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5-9

5-9

©G

lenc

oe/M

cGra

w-Hi

ll29

1G

lenc

oe A

lgeb

ra 2

Lesson 5-9

Pre-

Act

ivit

yH

ow d

o co

mp

lex

nu

mbe

rs a

pp

ly t

o p

olyn

omia

l eq

uat

ion

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 5-

9 at

the

top

of

page

270

in y

our

text

book

.

Supp

ose

the

num

ber

iis

defin

ed s

uch

that

i2

"!

1.C

ompl

ete

each

equ

atio

n.

2i2

"(2

i)2

"i4

"

Rea

din

g t

he

Less

on

1.C

ompl

ete

each

sta

tem

ent.

a.T

he f

orm

a%

biis

cal

led

the

of a

com

plex

num

ber.

b.In

the

com

plex

num

ber

4 %

5i,t

he r

eal p

art

is

and

the

imag

inar

y pa

rt is

.

Thi

s is

an

exam

ple

of a

com

plex

num

ber

that

is a

lso

a(n)

nu

mbe

r.

c.In

the

com

plex

num

ber

3,th

e re

al p

art

is

and

the

imag

inar

y pa

rt is

.

Thi

s is

exa

mpl

e of

com

plex

num

ber

that

is a

lso

a(n)

nu

mbe

r.

d.In

the

com

plex

num

ber

7i,t

he r

eal p

art

is

and

the

imag

inar

y pa

rt is

.

Thi

s is

an

exam

ple

of a

com

plex

num

ber

that

is a

lso

a(n)

nu

mbe

r.

2.G

ive

the

com

plex

con

juga

te o

f ea

ch n

umbe

r.

a.3

%7i

b.2

!i

3.W

hy a

re c

ompl

ex c

onju

gate

s us

ed in

div

idin

g co

mpl

ex n

umbe

rs?

The

prod

uct o

fco

mpl

ex c

onju

gate

s is

alw

ays

a re

al n

umbe

r.

4.E

xpla

in h

ow y

ou w

ould

use

com

plex

con

juga

tes

to f

ind

(3 %

7i) *

(2 !

i).

Writ

e th

edi

visi

on in

frac

tion

form

.The

n m

ultip

ly n

umer

ator

and

den

omin

ator

by

2 $

i.

Hel

pin

g Y

ou

Rem

emb

er

5.H

ow c

an y

ou u

se w

hat

you

know

abo

ut s

impl

ifyi

ng a

n ex

pres

sion

suc

h as

to

help

you

rem

embe

r ho

w t

o si

mpl

ify

frac

tion

s w

ith

imag

inar

y nu

mbe

rs in

the

deno

min

ator

?Sa

mpl

e an

swer

:In

both

cas

es,y

ou c

an m

ultip

ly th

enu

mer

ator

and

den

omin

ator

by

the

conj

ugat

e of

the

deno

min

ator

.

1 %

#3$

& 2 !

#5$

2 $

i

3 !

7i

pure

imag

inar

y 7

0re

al0

3im

agin

ary

54

stan

dard

form

1!

4!

2

©G

lenc

oe/M

cGra

w-Hi

ll29

2G

lenc

oe A

lgeb

ra 2

Conj

ugat

es a

nd A

bsol

ute

Valu

eW

hen

stud

ying

com

plex

num

bers

,it

is o

ften

con

veni

ent

to r

epre

sent

a c

ompl

ex

num

ber

by a

sin

gle

vari

able

.For

exa

mpl

e,w

e m

ight

let

z"

x%

yi.W

e de

note

th

e co

njug

ate

of z

by z $.

Thu

s,z $

"x

!yi

.

We

can

defi

ne t

he a

bsol

ute

valu

e of

a c

ompl

ex n

umbe

r as

fol

low

s.

-z-"

-x%

yi-"

#x2

%y

$2 $

The

re a

re m

any

impo

rtan

t re

lati

onsh

ips

invo

lvin

g co

njug

ates

and

abs

olut

e va

lues

of

com

plex

num

bers

.

Sh

ow +z

+2&

zz "fo

r an

y co

mp

lex

nu

mbe

r z.

Let

z"

x%

yi.T

hen,

z"

(x%

yi)(

x!

yi)

"x2

%y2

"#

( x2%

y2$

)2 $"

-z-2

Sh

ow

is t

he

mu

ltip

lica

tive

in

vers

e fo

r an

y n

onze

ro

com

ple

x n

um

ber

z.

We

know

-z-2

"zz $.

If z

# 0

,the

n w

e ha

ve z!

""1.

Thu

s,is

the

mul

tipl

icat

ive

inve

rse

of z

.

For

eac

h o

f th

e fo

llow

ing

com

ple

x n

um

bers

,fin

d t

he

abso

lute

val

ue

and

mu

ltip

lica

tive

in

vers

e.

1.2i

2;"! 2i "

2.!

4 !

3i5;

"!4 2$ 5

3i"

3.12

!5i

13;"12

1$ 695i

"

4.5

!12

i13

;"5$ 16

912i

"5.

1 %

i!

2";"1

! 2i

"6.

#3$

!i

2;

7.%

i8.

!i

9.&1 2&

!i

;1;

$i

1;"1 2"

$i

!3"

"3

!2"

"2

!2"

"2

!3"

!i!

3""

" 2!

6""

3

#3$

&2

#2$

&2

#2$

&2

#3$

&3

#3$

&3

!3"

$i

"4

z $& -z

-2

z $& -z

-2

z "" +z

+2

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD

____

_

5-9

5-9

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

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