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Prerequisite SkillsMasters

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ISBN: 0-07-820385-6 AMC Prerequisite Skills Masters

1 2 3 4 5 6 7 8 9 10 055 08 07 06 05 04 03 02 01 00

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Advanced Mathematical Concepts

ContentsPage

Pretest ....................................................................................................... 1Skill 1 Properties of Real Numbers ...................................................................... 3Skill 2 Fractions .................................................................................................... 5Skill 3 Order of Operations................................................................................... 7Skill 4 Algebraic Expressions ............................................................................... 9Skill 5 Solving Equations and Inequalities............................................................ 11Skill 6 Polynomials ............................................................................................... 13Skill 7 Operations with Polynomials ..................................................................... 15Skill 8 Rationalizing a Denominator ..................................................................... 17Skill 9 Factory by Grouping and Greatest Common Factor................................. 19Skill 10 Factoring Special Products of Polynomials............................................... 21Skill 11 Completing the Square.............................................................................. 23Skill 12 Statistical Displays..................................................................................... 25

Posttest...................................................................................................... 27Answers..................................................................................................... 29

Skill 1 Properties of Real NumbersName the property that justifies each statement.

1. If y � 2, then 5y � 5(2). 1.

2. (3 � 4) � 5 � 3 � (4 � 5) 2.

Skill 2 FractionsEvaluate each expression.

3. �156� � �

18

� 4. �164� � �

221� 3.

4.

5. �27

� � �56

� 6. �59

� � �23

� 5.

6.

Skill 3 Order of OperationsFind the value of each expression.

7. 2 � 4 � 3 – 5 7.

8. 4 – (3 � 5) � 2 8.

9. 15 � 2 � 32 – 6 � 4 9.

Skill 4 Algebraic Expressions10. Simplify 6r � 2rs – 5r – rs. 10.

11. Evaluate x � y2 – 4y if x � 3 and y � �2. 11.

Skill 5 Solving Equations and Inequalities12. Solve 4x � 6 � �3. 12.

13. Solve 7 � 3 – 8y. 13.

Skill 6 Polynomials14. Find the degree of 7x3 – xy. 14.

15. Arrange the terms of 7 – y3 � y so that the powers 15.of y are in descending order.

Skill 7 Operations with Polynomials16. Find (4m2 � 5m – 1) � (6m2 – 6m � 4). 16.

17. Find (5t � 3) – (3t � 4). 17.

18. Find (w � 5)(w – 2). 18.

Prerequisite Skills Pretest

NAME _____________________________ DATE ________________ PERIOD ________

© Glencoe/McGraw-Hill 1 Advanced Mathematical Concepts

Skill 8 Rationalizing a DenominatorSimplify.

19. 20. ��68

�� 21. 19.

20.

21.

Skill 9 Factoring by Grouping and Greatest Common FactorFactor each polynomial.22. 12xy – 8x2 22.

23. 2w – wv � 2r � rv 23.

Skill 10 Factoring Special Products of PolynomialsFactor.24. x2 – 16 24.

25. 4a2 – 25 25.

26. y2 – 8y � 16 26.

27. m2 � 6m � 9 27.

Skill 11 Completing the SquareFind the value of c that makes each trinomial a perfect square.28. x2 � 8x � c 28.

29. x2 – 16x � c 29.

Skill 12 Statistical Displays30. The line graph at the 30.

right shows the heightof a soybean plant recordedat the end of each week for 6 weeks. At the end of whichweek had the plant grownthe most?

Week0 1 2 3 4 5 6

0

1

2

Height(in.)

Soybean Growth

2�1 � �2�

�5��2�

Prerequisite Skills Pretest (continued)

NAME _____________________________ DATE ________________ PERIOD ________


Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL1


Properties of Real NumbersFor any real numbers a, b, and c, the following properties holdtrue.

PROPERTIES OF THE REAL NUMBERS Property Definition Example(s)

Closure Property a � b and ab are real numbers. 2 and 4 are real, so 2 � 4or 6 is real and 2 � 4 or 8 is real.

Commutative Property a � b � b � a and ab � ba 4 � 7 � 7 � 4; 5 � 8 � 8 � 5

Associative Property a � (b � c) � (a � b) � c 4 � (7 � 9) � (4 � 7) � 9; and a(bc) � (ab)c 3(6 � 5) � (3 � 6)5

Distributive Property a(b � c) � ab � ac 12(x � 4) � 12x � 48

Additive Identity a � 0 � a 3x � 0 � 3x

Multiplicative Identity a � 1 � a 7y � 1 � 7y

Additive Inverse For each real number a, there 3 � (�3) � 0 is a unique real number �asuch that a � (�a) � 0.

Multiplicative Inverse For each real number a 0, 7 � �17

� � 1 there is a unique real number �

1a

�

such that a � �1a

� � 1.

Multiplication Property a � 0 � 0 8x � 0 � 0 Of Zero

Reflexive Property a � a 9 � 9 Of Equality

Symmetric Property If a � b, then b � a. If 3 � x, then x � 3. Of Equality

Transitive Property If a � b and b � c, then a � c. If x � 2 and y � 2, Of Equality then x � y.

Substitution Property If a � b, then a and b may be If x � 2y and z � 3x,Of Equality substituted for one another in then z � 3(2y).

any expression involving a or b.

Trichotomy Property For any two real numbers x � 6, or x � 6, or x 6a and b, exactly one of the following is true: a � b, a � b, or a b.

Properties of Real Numbers

Complete each statement using the given property.1. 5(4 � 9) � , Associative Property

2. m � 0 � , Additive Identity

3. 6x � , Commutative Property

4. If 8 � 3n, then , Symmetric Property

5. 4(x – 6) � , Distributive Property

6. 8 � y � , Commutative Property

7. If t 7, then either or , Trichotomy Property

Name the property that justifies each statement.8. 3 is a real number. If x is a real number, then 3x is a real number.

9. If x � 7, then 6x � 6(7).

10. 9n � 1 � 9n

11. x � 8 and 8 � 2y, thus x � 2y

12. Given that x � 12, it must be that x � 12 or x � 12.

13. 4 � 72 � 4(70 � 2) � 4 � 70 � 4 � 2

14. 17 � 0 � 0

15. (5n � 7) � 4 � 5n � (7 � 4)

Prerequisite Skills Practice

NAME _____________________________ DATE ________________ PERIOD ________

SKILL1


FractionsFor any real numbers a, b, and c, the following rules and propertiesapply to fractions.

RULES FOR OPERATING ON AND PROPERTIES OF FRACTIONSRule or Property Definition Example(s)

Equivalent Fractions The same fraction represented �23

� and �69

� and are with different numbers. equivalent fractions. To write

equivalent fractions, multiply or divide the numeratorand denominator by the samenumber.

�23

�

�

33

� � �69

� �69

�

�

33

� � �32

�

Adding and Subtracting For fractions �ac

� and �bc

�, �19

� � �29

� � �39

� or �31

�

Like Fractions where c � 0,�1145� � �

3125� � ��

1158� or �1�

51

��ac

� � �bc

� � �a �

cb

� and

�ac

� � �bc

� � �a �

cb

�.

Adding and Subtracting To find the sum or difference Unlike Fractions of two fractions with unlike

denominators, rename the fractions with a common denominator. Then add or subtract and simplify.

Multiplying Fractions For fractions �ab

� and �dc

�, �152� � �

29

� � �152

�

�

29

� � �11008

� or �554�

where b ≠ 0 and d ≠ 0, or

�ab

� � �dc

� � �ba

dc� �

152� � �

29

� � �152� � �

29

� or �554�

Dividing Fractions For fractions �ab

� and �dc

�, �3�37� � �

67� � ��

274� � �

67� �

where b ≠ 0 and d ≠ 0, ��274� � �

76� � �4

�ab

� � �dc

� � �ab

� � �cd

�

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL2


�79

� � ��1112� � �

79

� � ��44

� � �1112� � �

33

�

� �2386� � ��

333

6�

� �6316� or 1�

3265�

�14

� � �23

� � �14

� � �33

� � �23

� � �44

�

� �132� � �

182� or � �

152

�

1

6

4 1

1 1

Fractions

Evaluate each expression.1. �

57

� � �17

� 2. �1290� � �

230� 3. ��

245� � �

235�

4. �57

� � �67

� 5. 2�158� � �

138� 6. ��

176� � �

1165�

7. �172� � �

16

� 8. 4�59

� � �19

� 9. �170� � 2�

145�

10. 2�19

� � �152� 11. �14�

13

� � 2�16

� 12. �274� � �

110�

13. 1�114� � �

47

� 14. �258� � 1�

134� 15. 1�

270� � 2�

18

�

16. �57

� � �17

� 17. �14

� � �89

� 18. �134� � 1�

13

�

19. �251� � ��

175�� 20. ��

172�� 3�

134� 21. �

156� � �

58

�

22. �57

� � �17

� 23. �112� � �

14

� 24. ��1�25

�� 134�

25. �145� � ��

29

�� 26. 2�118� � 1�

56

� 27. 4�56

� � ��23

��


NAME _____________________________ DATE ________________ PERIOD ________

SKILL2


Order of OperationsThe order for performing operations is as follows.

1. Simplify the expressions inside grouping symbols, such as parentheses,brackets, braces, and fraction bars.

2. Evaluate all powers.

3. Do all multiplications and divisions from left to right.

4. Do all additions and subtractions from left to right.

Example 1 Find the value of (4 � 2)2 � 15 � 3.

(4 + 2)2 – 15 � 3 � 62 � 15 � 3 First add 4 and 2.

� 36 � 15 � 3 Evaluate the power.

� 36 � 5 Divide 15 by 3.

� 31 Subtract 5 from 36.

Example 2 Find the value of �32(4

��

42

2

)�.

�32(4

��

42

2

)� � �

23�(2

4)

2� Subtract 2 from 4.

� �2

3�(2

1)6

� Evaluate the power.

� �2 �

616� Multiply 3 and 2.

� �168� Add 2 and 16.

� 3 Divide 18 by 6.

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL3


Order of Operations

Find the value of each expression.1. 2(42 � 3) 2. 3(4 � 5) � 2

3. 7 � 22 � 5(3) 4. 5 � [(3 � 4) � 1]

5. 12 � 14 � 2 � 5 6. [(�2 � 4) � 5] � 2

7. 6 � [28 � (3 � 4)] 8. 0.5(23 � 4)

9. �5(62 � 4) � 15 10. �12(3

32� 8)�

11. (4 � 5 � 3) � 4 � 2 12. 7 � 15 � 3 � 4 � 3

13. �27

�� 4

5��

31

� 14. 5 � (6 � 2) � 2

15. 1 � 4(24 � 1) � 6 16. 8 � 22 � 4 � 5 � 3

17. 32 � 6 � 2 � 8 18. 72 � [4 � 3(6 � 2) � 1]

19. 5 � [6 � 3(4 � 3) � 3] 20. 1 � 3 � 2 � 3 � 2


NAME _____________________________ DATE ________________ PERIOD ________

SKILL3


Algebraic ExpressionsAn algebraic expression consists of one or more numbers andvariables along with one or more arithmetic operations.Some examples of algebraic expressions are given below.

a � 3 4x 9 � c �4y

� 4x � 6 9mn � p

It is often necessary to translate verbal expressions into algebraicexpressions.

Verbal Expression Algebraic Expression

5 less than the product of 4 and a number n 4n � 5

The quotient of two more than a number and 3 (x � 2) � 3

Four more than twice r 4 � 2r

Sometimes algebraic expressions need to be simplified.An algebraic expression is in simplest form when it contains no liketerms and no parentheses or brackets.

Algebraic Expression Simplest Form

4x � 3x 4x and 3x are like terms because theycontain the same variable. To simplify,add the coefficients. So 4x � 3x � 7x

5(x � y2) � 2y2 Distribute the 5 to x and y2. Thencombine like terms.5(x � y2) � 2y2 � 5x � 5y2 � 2y2

� 5x � 3y2

To evaluate an algebraic expression, substitute given values for thevariables and use the order of operations to evaluate.

Algebraic Expression Value of Expression if x � 4 and y � �2

2x � 4y 2x � 4y � 2(4) � 4(�2) � 8 � 8 or 16

�xy

� � y2 �xy

� � y2 � ��42� � ( �2)2

� �2 � 4 or �6

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL4


Algebraic Expressions

Write an algebraic expression for each verbal expression.1. The sum of three and the square of x

2. Twice the difference of 5 and y

3. The product of 5 and 3 more than the product of m and n

4. Half the sum of a and the square of b

Simplify.5. 6y � 2x � 4y 6. 6(m � n) � 4n

7. 2s2 � 2t � 4t � s2 8. 8ab � 7cd � 2(ab � cd)

9. 4(2y � x) � 2( y � 2x) 10. 4m � 2(m2 � n) � 5n

11. w2 � 4w � 4w � 3w2 12. 8mn � 7mn2 � (mn2 � pq)

Evaluate if a � 5 and b � �3.13. 4a � 3b 14. a2 � b2

15. a � 3(a � b) 16. 5a2 � 4(3b � 2)

17. (5a)2 � 5b 18. 14a � 7b � 2(a � b)

19. 6a � (a � 4b) 20. 10a2 � 2(b � 1)


NAME _____________________________ DATE ________________ PERIOD ________

SKILL4


Solving Equations and InequalitiesTo solve an equation or an inequality means to isolate the variablethat has a coefficient of 1 on one side of the equation or inequality.

Example 1 Solve 14 � 4t � 16 � 2t � 2.

14 � 4t � 16 � 2t � 2

14 � 4t � 18 � 2t Simplify each side.

14 � 4t � 2t � 18 � 2t � 2t Add 2t to each side.

14 � 2t � 18

14 � 14 � 2t � 18 � 14 Subtract 14 from each side.

�2t � 4

��

�

22t

� � ��

42� Divide each side by �2.

t � �2 Simplify.

Since t is now isolated, �2 is the solution to the equation. To checkthis solution, substitute �2 for t in the original equation.

Check: 14 � 4t � 16 � 2t � 2

14 � 4(�2) � 16 � 2(�2) � 2

14 � 8 � 16 � 4 � 2

22 � 22 ✓

Example 2 Solve 5 � �6 � �4z

�.

5 � �6 � �4z�

5 � 6 � �6 � 6 � �4z

� Add 6 to each side.

11 � ��4z

� Simplify.

4(11) � 4��4z

�� Multiply each side by 4.

44 � �z Simplify.

�44 z Multiply each side by �1. When multiplying or dividing eachside of an inequality by a negative number, the direction of theinequality sign must be changed.

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL5


Solving Equations and Inequalities

Solve.1. 2x � 3 � 7 2. 7 � 4y � �5

3. 3 � �m4� � 6 4. �

23

� r � 7 � 2

5. 12 � (x � 1) � 3 � x 6. 2(1 � 4y) � 2y � �20 � 2y

7. 10m � 5 � 6m � 3 8. 4r � 5 � 7r � 7

9. 2x � 2 � 4 10. �1 � 8y � 7

11. 5x � 2 � 4x � 16 12. 6 � �4z

� � 2 1

13. 2(x � 3) � 5 10 � x 14. 2 � 5(a � 3) � 2

15. 4 � 6w � 6 � 2(w � 5) 16. 18 � 4(3 � 2s) � 4


NAME _____________________________ DATE ________________ PERIOD ________

SKILL5


PolynomialsA polynomial is an algebraic expression that is the sum of one, two, three, or manyterms. A monomial is a polynomial with one term and is a number, a variable, or aproduct of numbers and variables. The exponents of the variables of a monomialmust be positive. A monomial that is a number is called a constant. A binomial isthe sum of two monomials. A trinomial is the sum of three monomials. Here aresome examples of each.

Monomial Binomial Trinomial

2y 3y � 1 4xy � 2y � z

4abc2 4m � 2n 3a2 � 4ab4 � 4

5 5x2 � 10xy 6m4 � 5mn � 2n3

�2x3 16 � 2d3 3 � 7w � z2

The degree of a monomial is the sum of the exponents of its variables.

Monomial Degree

2y 1

4abc2 1 � 1 � 2 � 4

5 0

�2x3 3

To find the degree of a polynomial, you must find the degree of each monomial. Thegreatest degree of any monomial is the degree of the polynomial.

Degree of Degree ofTerms Polynomial

4m � 2n 4m, 2n 1, 1 1

3a2 � 4ab4 � 4 3a2, 4ab4, 4 2, 5, 0 5

6m4 � 5mn � 2n3 6m4, 5mn, 2n3 4, 2, 3 4

3 � 7w � z2 3, 7w, z2 0, 1, 2 2

The terms of a polynomial are usually arranged so that the powers of one variableare in ascending or descending order. For the polynomial 3b2 � 4b4 � 4, ascendingorder is 4 � 3b2 � 4b4 and descending order is 4b4 � 3b2 � 4.

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL6


Polynomial Terms

Polynomials

State whether each expression is a polynomial. If the expressionis a polynomial, identify it as a monomial, a binomial,or a trinomial.1. 5x2 � 2 2. �

x52�

3. 0.2y � y3 � 5 4. �12

�v3 � �13

�v

5. 8xy6 6. 17fg3 � fg � 3

Find the degree of each polynomial.7. 6x2 � xy 8. 15r3t4

9. 4 � r � s � rs � t4 10. 6x2y3z � 8x3y4z2

11. 9w3 � 4wy4 � 6y2 12. 2h � 7 � 8h5 � 7h2

Arrange the terms of each polynomial so that the powers of xare in descending order.13. 6 � x2 � x 14. 7x2 � x3y � x � 1

15. 8x4z � 7x5 � 4x2 � x 16. 4x � 5x3y � x2

Arrange the terms of each polynomial so that the powers of xare in ascending order.17. 5 � 2x2 � 4x 18. 7x2 � x3y � x4 � 4

19. x4z � 7x8 � 2x � x2 20. 2x5 � 5x3y � x4


NAME _____________________________ DATE ________________ PERIOD ________

SKILL6


Operations with PolynomialsTo add polynomials, group like terms and then find the sum.

Example 1

Find (3a2 � 5a � 10) � (6a2 � 8a � 2).

(3a2 � 5a – 10) � (6a2 – 8a � 2) � (3a2 � 6a2) � (5a – 8a) � (�10 � 2)

� 9a2 – 3a – 8

To subtract polynomials, add the additive inverse.

Example 2

Find (2y2 � 3y � 6) � (5y2 � 9y � 4).

The additive inverse of 5y2 � 9y – 4 is �5y2 � 9y � 4.

Add this polynomial to 2y2 � 3y � 6.

(2y2 � 3y � 6) � (5y2 � 9y � 4) � (2y2 � 3y � 6) � (�5y2 � 9y � 4)

� (2y2 � 5y2) � (�3y – 9y) � (6 � 4)

� �3y2 – 12y � 10

To multiply polynomials, use the FOIL method which is describedbelow.

To multiply two binomials, find the sum of the products of

F the First terms,

O the Outer terms,

I the Inner terms, and

L the Last terms.

Example 3

Find (y � 3)(2y � 5).

F O I L( y � 3)(2y � 5) � ( y)(2y) � ( y)(�5) � (3)(2y) � (3)(�5)

� 2y2 – 5y � 6y � 15

� 2y2 � y � 15

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL7


Operations with Polynomials

Find each sum.1. (2a2 � 4a � 1) � (a2 – 6a � 4) 2. (6m2 � 4m – 6) � (6m � 7)

3. (�2r2 � 5r – 4) � (�r2 � 5r) 4. (7t3 � 3t – 3) � (3t2 – 8t � 6)

5. (9w – 15) � (�10w2 � 6w) 6. (4c4 � 3c3 – 2c) � (�c4 � 7c3 � 2)

7. (2d � 6) � (�4d3 � 4d � 3) 8. (2v3 � 5v2 – 5) � (3v2 � 3v � 7)

Find each difference.9. (2a2 � 4a � 1) � (a2 – 6a � 4) 10. (6m2 � 4m – 6) � (6m � 7)

11. (�2r2 � 5r – 4) � (�r2 � 5r) 12. (7t3 � 3t – 3) � (3t2 – 8t � 6)

13. (9w – 15) � (�10w2 � 6w) 14. (4c4 � 3c3 – 2c) � (�c4 � 7c3 � 2)

15. (2d � 6) � (�4d3 � 4d � 3) 16. (2v3 � 5v2 – 5) � (3v2 � 3v � 7)

17. (4w4 – 10) � (�3w4 – 5w – 1) 18. (3g3 � 3g2 – 2g) � (g3 – 5g � 5)

Find each product.19. (4a � 1)(a � 4) 20. (4m – 5)(6m � 1)

21. (5r – 3)(r � 4) 22. (t2 � 3)(3t2 – 1)

23. (2w – 5)(w2 � 6) 24. (3c2 – 2)(c � 2)

25. (9x – 5)(�10x2 � 7) 26. (4c � 3d)(c � 4d)

27. (2m � 5n)(8m – 6n) 28. (2f � 3gh)(4f � gh)


NAME _____________________________ DATE ________________ PERIOD ________

SKILL7


Rationalizing a DenominatorWhen you rationalize a denominator, you remove or eliminate radicals from thedenominator of a fraction. The denominator of a fraction must be rationalized for it to bein simplest form.

Example 1 Simplify .

� � � 1

� The denominator is now a rational number.So the denominator has been rationalized.

Example 2 Simplify ��.

��1182�� Simplify the radicals in the numerator

and denominator.

� � � 1

� or

Binomials of the form a�b� � c�d� and a�b� � c�d� are called conjugates.

Conjugates are often used to rationalize the denominators of fractions containingradicals because their product is always a rational number with no radicals.

Example 3 Simplify .

� � 4 � �3� and 4 � �3� are conjugates.

� or 16 � 4�3��

1316 � 4�3��

16 � 3

4 � �3��4 � �3�

4�4 � �3�

4�4 � �3�

4�4 � �3�

�6��

2

3�6��

6

�3��3�

�3��3�

3�2��2�3�

3�2��2�3�

18�12

�1�0��2

�2��2�

�2��2�

�5��2�

�5��2�

�5��

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL8


Rationalizing a Denominator

Simplify.

1. 2. 3. ��2342��

4. 5. 6. ��490��

7. 8. 9. 6��2402��

10. 11. 12.

13. 14. 15.

16. 17. ��2306�� 18. 10��

190��

19. 20. 21.

22. 23. 24.3�2��1 � �2�

7��1 � 4�5�

6��1 � 2�2�

4�5 � �5�

3�2 � �2�

1�1 � �3�

3�28��

�3�

12�12��

8�8�6�40��

�5��7��5�

8�2��9�3�

5�2��6�8�

8�7��

24

3��28�

6��12�

3�15��20�

3�2��14�

�8��3�

1��3�


NAME _____________________________ DATE ________________ PERIOD ________

SKILL8


Factoring by Grouping andGreatest Common Factor

The greatest common factor (GCF) of two or more monomials is the product of theircommon factors when each monomial is expressed in factored form. A polynomial cansometimes be factored by using the Distributive Property to factor out the greatestcommon factor of the terms in the polynomial.

Example 1

Factor 12mn2 � 18m2n2.

First find the GCF of 12mn2 and 18m2n2.

12mn2 � 2 � 2 � 3 � m � n � n

18m2n2 � 2 � 3 � 3 � m � m � n � n

The GCF is 2 � 3 � m � n � n � 6mn2.

12mn2 � 18m2n2 � 6mn2(2) � 6mn2(3m) Write each term as the product of the GCF and another factor.

� 6mn2(2 – 3m) Use the Distributive Property.

A factoring method called factoring by grouping can be used to factor a four-termpolynomial into its two binomial factors. This method is illustrated in the followingexample.

Example 2

Factor 12ac � 21ad � 8bc � 14bd.

12ac � 21ad � 8bc � 14bd

� (12ac � 21ad) � (8bc � 14bd) Group the terms so that each group has a GCF other than 1.

� 3a(4c � 7d) � 2b(4c � 7d) Factor out the GCF from each group.� (3a � 2b)(4c � 7d) (4c � 7d) is the common factor.

Use the Distributive Property to factor.

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL9


Factoring by Grouping andGreatest Common Factor

Factor each polynomial.1. 9x2 � 36x 2. 14ab – 18ab2

3. 15mn3 � n4 4. 17c – 41c2d

5. r2 � 6rs 6. 7h – 14h2k

7. 2ax � 6xc � ba � 3bc 8. 2my � 7x � 7m � 2xy

9. 3m2 � 5m2p � 3p2 � 5p3 10. 3x3y � 9xy2 � 36xy

11. 12ax � 20bx � 32cx 12. 4ax – 14bx � 35by – 10ay

13. 5a2 – 4ab � 12b3 – 15ab2 14. 2x3 – 5xy2 – 2x2y � 5y3

15. 3a3 � 2ab – 15a2 – 10b 16. 3x2 – 2xy � 10y – 15x

17. m2 – mn – 7n � 7m 18. ay – ab � cb � cy

19. rx � 2ry � kx � 2ky 20. 5a – 10a2 � 2b – 4ab


NAME _____________________________ DATE ________________ PERIOD ________

SKILL9


Factoring Special Products of Polynomials

FACTORING SPECIAL PRODUCTS OF POLYNOMIALS

Name of Polynomial Definition Example(s)

Difference a2 � b2 � (a – b)(a � b) of Squares � (a � b)(a � b)

x 2 – 4 � x 2 – 2 2

� (x – 2)(x � 2)� (x � 2)(x – 2)

9m 2 – 25y 4 � (3m) 2 – (5y 2) 2

� (3m � 5y 2)(3m � 5y 2)� (3m � 5y 2)(3m � 5y 2)

Perfect Square The expansion of the To determine if x 2 � 6x � 9 is aTrinomial square of a sum or the perfect square trinomial, answer

square of a difference. the following questions.

Square of a sum: • Is the first term a perfect square?(a � b) 2 � a 2 � 2ab �b 2 yes

• Is the last term a perfect square?Square of a difference: yes(a �b) 2 � a 2 � 2ab �b 2 • Is the middle term twice the product

of the square root of the first termand the square root of the last term?In other words, does 6x � 2(x)(3)?yes

x 2 � 6x � 9 is a perfect square trinomial.

x 2 � 6x � 9 � x 2 � 2(x)(3) � 3 2

� (x � 3) 2

Likewise, 4m 2 – 4m � 1 is a perfect square trinomial. It can be factored into the square of a difference.

4m 2 – 4m � 1

� (2m) 2 – 2(2m)(1) � 12

� (2m – 1) 2

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL10


Factoring Special Products of Polynomials

Factor.1. x2 � 25 2. y2 � 81

3. 4x2 � 1 4. a2 � 144

5. 16m2 � 9 6. 36y2 � 1

7. 64 � x2 8. 0.25y2 � 4

9. x2 � 10x � 25 10. y2 � 18y � 81

11. r2 � 8r � 16 12. w2 – 20w � 100

13. 4x2 � 4x � 1 14. 16y2 � 24y � 9

15. 25s2 � 20s � 4 16. 36t2 � 12t � 1

17. 9j2 – 30j � 25 18. 9h2 – 42h � 49

19. 49 – 14x � x2 20. 16 � 56y � 49y2

21. 4x2 � 12xy � 9y2 22. 25c2 – 30cd � 9d2


NAME _____________________________ DATE ________________ PERIOD ________

SKILL10


Completing the SquareTo complete the square for a quadratic expression of the form x2 � bx, you can follow thesteps below.

Step 1 Find �12

� of b, the coefficient of x.

Step 2 Square the result of Step 1.

Step 3 Add the result of Step 2 to x2 � bx, the original expression.

Example 1

Find the value of c that makes the trinomial x2 � 10x � ca perfect square.

Step 1 Find �12

� of 10. �120� � 5

Step 2 Square the result of Step 1. 52 � 25

Step 3 Add the result of Step 2 to x2 � 10x. x2 � 10x � 25

Thus, c � 25. Notice that x2 � 10x � 25 � (x � 5)2.

Example 2

Find the value of c that makes the trinomial 2x2 � 14x � ca perfect square.

Step 1 Find �12

� of 14. �124� � 7

Step 2 Square the result of Step 1. 72 � 49

Step 3 Since the coefficient of x2 is not 1, c � �429� or 24.5

you must divide the result of Step 2 by the coefficient of x2 to find c.

Step 4 Add the result of Step 3 to 2x2 � 14x. 2x2 � 14x � 24.5

Thus c � 24.5. Notice that 2x2 � 14x � 24.5 � 2(x � 3.5)2.

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL11


Completing the Square

Find the value of c that makes each trinomial a perfect square.1. x2 � 6x � c 2. x2 � 14x � c

3. x2 � 16x � c 4. x2 � 24x � c

5. x2 – 40x � c 6. x2 � 2x � c

7. x2 � 3x � c 8. x2 � 11x � c

9. x2 � 25x � c 10. x2 � 13x � c

11. x2 � 16x � c 12. x2 � 30x � c

13. x2 � 100x � c 14. x2 � 80x � c

15. x2 � 45x � c 16. x2 � 36x � c

17. x2 � 19x � c 18. x2 – 43x � c

19. 4x2 � 16x � c 20. 9x2 � 2x � c

21. 2x2 � 18x � c 22. 16x2 � 16x � c

23. 7x2 � 14x � c 24. 25x2 � 10x � c

25. 8x2 � 24x � c 26. 12x2 � 12x � c

27. 15x2 � 30x � c 28. 5x2 � 20x � c

29. 12x2 � 36x � c 30. 4x2 � 19x � c

31. 5x2 � 30x � c 32. 13x2 � x � c


NAME _____________________________ DATE ________________ PERIOD ________

SKILL11


Statistical DisplaysOne way statistical information can be displayed is in a bar graph.

Example 1

The manager of a movie theateris studying the number of ticketssold each day during one week. He created the following table.Create a bar graph of this data.

Each of the categories in a bar graph has a bar to represent it.The vertical scale shows the number of tickets sold.

Line graphs can also be used to display statistical information. These graphs areusually used to show change over a period of time.

Example 2

The line graph at the right showsthe temperatures recorded one dayin March. During which hour wasthe change the greatest? the least?

The greatest change occurred where theslope of the segment is the greatest,between 10 A.M. and 11 A.M.The least change occurred where the slope of the segment is the least, between8 A.M. and 9 A.M.

Prerequisite Skills

NAME _____________________________ DATE ________________ PERIOD ________

SKILL12


Number ofTickets Sold

Sunday 263Monday 121Tuesday 100

Wednesday 55Thursday 92

Friday 193Saturday 300

DayS M T W T F S

0

300

200

100

TicketsSold

Time (A.M.)6 7 8 9 10 110

60

50

40

20

10

30

Temperature(˚F)

Day

Statistical Displays1. Create a bar graph to show the type of movies the tickets were sold for on Saturday.

For which types of movies were about the same number of tickets sold?

Type of Movie Number of Tickets Sold

Comedy 79

Action/Adventure 28

Drama 121

Horror 46

Other 26

2. Create a line graph to show the amount of precipitation last week. Between whichtwo days was the increase in precipitation the greatest?

Amount of Precipitation( inches)

Sunday 0.4

Monday 2

Tuesday 1.8

Wednesday 2.9

Thursday 2.5

Friday 0.6

Saturday 0.5


NAME _____________________________ DATE ________________ PERIOD ________

SKILL12


Day

Skill 1 Properties of Real NumbersName the property that justifies each statement.

1. 8r �1 � 8r 1.

2. 7mn � 7nm 2.

Skill 2 FractionsEvaluate each expression.

3. �241� � �

134� 4. �

158� � �

176� 3.

4.

5. �59

� � �175� 6. �

152� � �

34

� 5.

6.

Skill 3 Order of OperationsFind the value of each expression.

7. 6 � 7 � 2 � 1 7.

8. 8 � (4 � 6) � 2 8.

9. 12 � 3 � 22 � 3 � 5 9.

Skill 4 Algebraic Expressions10. Simplify 5f � 2fg – 4f – fg. 10.

11. Evaluate w � z3 – 3z if w � 5 and z � �4. 11.

Skill 5 Solving Equations and Inequalities12. Solve 3x � 7 � 5. 12.

13. Solve 16 6 � 5y. 13.

Skill 6 Polynomials14. Find the degree of 5x4 – x5y. 14.

15. Arrange the terms of 5 – t2 � t so that the powers 15.of t are in descending order.

Prerequisite Skills Posttest

NAME _____________________________ DATE ________________ PERIOD ________


Skill 7 Operations with Polynomials16. Find (6g2 � 4g – 15) � ( g2 – 6g � 4). 16.

17. Find (8w � 3) – (2w � 1). 17.

18. Find (v � 1)(v – 8). 18.

Skill 8 Rationalizing a DenominatorSimplify.

19. 20. ��1125�� 21. 19.

20.

21.

Skill 9 Factoring by Grouping and Greatest Common FactorFactor each polynomial.22. 5k2 � 15hk 23. 21n � 7np � 3m � mp 22.

23.

Skill 10 Factoring Special Products of PolynomialsFactor.24. x2 � 49 25. 9a2 � 1 24.

25.

26. y2 � 10y � 25 27. m2 � 20m � 100 26.

27.

Skill 11 Completing the SquareFind the value of c that makes each trinomial a perfect square.28. x2 � 4x � c 29. x2 � 34x � c 28.

29.

Skill 12 Statistical Displays30. The line graph at the 30.

right shows the lengthof a baby recorded at eachmonth of age for 6 monthsDuring which time perioddid the baby grow the most?

4�2 � �2�

�5��3�

Prerequisite Skills Posttest (continued)

NAME _____________________________ DATE ________________ PERIOD ________


Month0 1 2 3 4 5 6

0

10

30

20Height(in.)

Page 1 Page 2

1. Substiution Property 19. ��12�0��

2. Associative Property of Addition 20. ��23

��3. �

176� 21. �2 � 2�2�

4. �13

� 22. 4x(3y � 2x)

5. �251� 23. (w � r)(2 � v)

6. �56

� 24. (x � 4)(x � 4)

7. 9 25. (2a � 5)(2a � 5)

8. 0 26. (v � 4)2

9. 65.5 27. (m � 3)2

10. r � rs 28. 16

11. 15 29. 64

12. � 2.25 30. 6

13. y � �0.5

14. 3

15. � y3 � y � 7

16. 10m2 � m � 3

17. 2t � 1

18. w2 � 3w � 10

Prerequisite Skills Pretest Answers


Skill 1, Page 4 Skill 2, Page 6

1. (5 � 4) � 9 1. �67

� 14. �1�218�

2. m 2. 1�110� 15. ��

4301�

3. x � 6 3. ��215� 16. �

459�

4. 3n � 8 4. ��17

� 17. �92

�

5. 4x � 4 � 6 5. 2�19

� 18. �72

�

6. y � 8 6. �1�38

� 19. ��91

�

7. t 7, t � 7 7. �34

� 20. �1�87

�

8. Closure Property 8. 4�23

� 21. �12258

�

9. Substitution Property 9. 2�2390� 22. 5

10. Multiplicative Identity 10. 2�1396� 23. �

31

�

11. Transitive Property 11. �12�16

� 24. �6�185�

12. Trichotomy Property 12. �12230

� 25. �1�51

�

13. Distributive Property 13. �12

� 26. 1�343�

14. Multiplication Property of Zero 27. �7�14

�

15. Associative Property

Prerequisite Skills Answers



1. 26 1. 3 � x2

2. 25 2. 2(5 � y)

3. 18 3. 5(mn � 3)

4. �1 4. 0.5(a � b2)

5. 24 5. 2y � 2x

6. �0.5 6. 6m � 2n

7. �55 7. s2 � 2t

8. 2 8. 6ab � 9cd

9. �60 9. 6y

10. 14 �23

� 10. 4m � 2m2 � 3n

11. 21 11. �2w2 � 8w

12. �10 12. 8mn � 8mn2 � pq

13. 4�16

� 13. 29

14. �11 14. 34

15. �407 15. 11

16. �9 16. 81

17. 20 17. 640

18. 10 18. 53

19. �40 19. 13

20. 1 20. 254




1. 2 1. yes; binomial

2. 3 2. no

3. �12 3. yes; trinomial

4. �7.5 4. yes; binomial

5. 4 5. yes; monomial

6. 2.75 6. yes; trinomial

7. 0.5 7. 2

8. ��23

� 8. 7

9. x 1 9. 4

10. y �1 10. 9

11. x � 2 11. 5

12. z � 28 12. 5

13. x 9 13. x2 � x � 6

14. a 3 14. x3 � 7x2 � x � 1

15. w 2.5 15. �7x5 � 8x4z � 4x2 � x

16. s � �1.25 16. 5x3y � x2 � 4x

17. 5 � 4x � 2x2

18. 4 � 7x2 � x3z � x4

19. 2x � x2 � x4z � 7x8

20. 5x3y � x4 � 2x5




1. 3a2 � 2a � 3 1. ��

33��

2. 6m2 � 2m � 13 2. �2�

36��

3. �3r2 � 10r � 4 3. ��

23��

4. 7t3 � 3t2 � 5t � 3 4. �3�

77��

5. �10w2 � 3w � 15 5. �3�

23��

6. 3c4 � 10c3 � 2c � 2 6. �3�

201�0��

7. �4d3 � 6d � 9 7. �3�

8. 2v3 � 8v2 � 3v � 12 8. �31�

47��

9. a2 � 10a � 5 9. �6�

22�11�0��

10. 6m2 � 10m � 1 10. �2�

34�2��

11. �r2 � 4 11. �152�

12. 7t3 � 3t2 � 11t � 9 12. 86\27

13. 10w2 � 15w � 16 13. ��

53�5��

14. 5c4 � 4c3 � 2c � 2 14. 12�2�

15. 4d3 � 2d � 3 15. �3�

46��

16. 2v3 � 2v2 � 2 16. 2�2�1�

17. 7w4 � 5w � 9 17. ��

35��

18. 2g3 � 3g2 � 3g � 5 18. 3�1�0�

19. 4a2 � 15a � 4 19. ��1 �

2�3��

20. 24m2 � 34m � 5 20. �6 �

23�2��

21. 5r2 � 17r � 12 21. �5 �

5�5��

22. 3t4 � 8t2 � 3 22. ��6 �

712�2��

23. 2w3 � 5w2 � 12w � 30 23. �� 7 �

7928�5��

24. 3c3 � 6c2 � 2c � 4 24. �3�2� �� 6�

25. �90x3 � 50x2 � 63x � 35

26. 4c2 � 19cd � 12d2

27. 16m2 � 28mn � 30n2

28. 8f 2 � 10fgh � 3g2h2




1. 9x(x � 4) 1. (x � 5)(x � 5)

2. 2ab(7 � 9b) 2. (y � 9)( y � 9)

3. n3(15m � n) 3. (2x � 1)(2x � 1)

4. c(17 � 41cd) 4. (a � 12)(a � 12)

5. r(r � 6s) 5. (4m � 3)(4m � 3)

6. 7h(1 � 2hk) 6. (6y � 1)(6y � 1)

7. (2x � b)(a � 3c) 7. (8 � x)(8 � x)

8. (m � x)(2y � 7) 8. (0.5y � 2)(0.5y � 2)

9. (m2 � p2)(3 � 5p) 9. (x � 5)2

10. 3xy(x2 � 3y � 12) 10. (y � 9)2

11. 4x(3a � 5b � 8c) 11. (r � 4)2

12. (2x � 5y)(2a � 7b) 12. (w � 10)2

13. (a � 3b2)(5a � 4b) 13. (2x � 1)2

14. (2x2 � 5y2)(x � y) 14. (4y � 3)2

15. (3a2 � 2b)(a � 5) 15. (5s � 2)2

16. (x � 5)(3x � 2y) 16. (6t � 1)2

17. (m � n)(m � 7) 17. (3j � 5)2

18. (a � c)(y � b) 18. (3h � 7)2

19. (r � k)(x � 2y) 19. (7 � x)2

20. (5a � 2b)(1 � 2a) 20. (4 � 7y)2

21. (2x � 3y)2

22. (5c � 3d)2




1. 9 1. Action/Adventure and Other

2. 49

3. 64

4. 144

5. 400

6. 1

7. 2.25

8. 30.25

9. 156.25

10. 42.25

11. 64

12. 225

13. 2500

14. 1600 2. Tuesday and Wednesday

15. 506.25

16. 324

17. 90.25

18. 462.25

19. 16

20. �19

�

21. 40.5

22. 4

23. 7

24. 1

25. 18

26. 3

27. 15

28. 20

29. 27

30. 22.5625

31. 45

32. �512�



Comed

y

Action

/

Adven

ture

Drama

Horror

Other0

120

100

40

60

20

80Number

ofTicketsSold

Type of Movie

DayS M T W T F S

0

1

3

2

0.5

2.5

1.5Precipitation

(in.)

Prerequisite Skills Posttest Answers


Page 27 Page 28

1. Multiplicative Identity 16. 7g2 � 2g � 8

2. Commutative Property 17. 6w � 2

3. �1472� 18. v2 � 7v � 8

4. ��12434

� 19. ��13�5��

5. �277� 20. �

255�

6. �59

� 21. 4 � 2�2�

7. 19 22. 5k(k � 3h)

8. 9 23. (7n � m)(3 � p)

9. 18 24. (x � 7)(x � 7)

10. 9f � fg 25. (3a � 1)(3a � 1)

11. �47 26. (y � 5)2

12. 4 27. (m � 10)2

13. y � 2 28. 4

14. 6 29. 289

15. � t2 � t � 5 30. between 3rd and 4th month

Documents

Prerequisite Skills Masters - wikispaces.netadvancedfunctions.cmswiki.wikispaces.net/file/view/prerequisite... · Skill 8 Rationalizing a Denominator..... 17 Skill 9 Factory by Grouping