Dividing Polynomials

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Name Class Date

Explore Evaluating a Polynomial Function Using Synthetic Substitution

Polynomials can be written in something called nested form. A polynomial in nested form is written in such a way

that evaluating it involves an alternating sequence of additions and multiplications. For instance, the nested form of

p (x) = 4 x 3 + 3 x 2 + 2x + 1 is p (x) = x (x (4x + 3) + 2) + 1, which you evaluate by starting with 4, multiplying by

the value of x, adding 3, multiplying by x, adding 2, multiplying by x, and adding 1.

Given p (x) = 4 x3 + 3 x2 + 2x + 1, find p (-2) .

You can set up an array of numbers that captures the sequence of multiplications and additions needed to find p (a). Using this array to find p (a) is called synthetic substitution.

Given p (x) = 4 x 3 + 3 x 2 + 2x + 1, find p (-2) by using synthetic substitution. The dashed arrow indicates bringing down, the diagonal arrows represent multiplication by –2, and the solid down arrows indicate adding.

The first two steps are to bring down the leading number, 4, then multiply by the value you are evaluating at, -2.

Add 3 and –8.

Resource Locker

4

4

3

-8

2 1-2

4

4

3

-8

2 1-2

-5

Rewrite p (x) as p (x) = x (x (4x + 3) + 2) + 1.

Multiply. -2 · 4 = -8

Add. -8 + 3 = -5

Multiply. -5 · (-2) = 10

Add. 10 + 2 = 12

Multiply. 12 · (-2) = -24

Add. -24 + 1 = -23

Module 6 321 Lesson 5

6 . 5 Dividing PolynomialsEssential Question: What are some ways to divide polynomials, and how do you know when

the divisor is a factor of the dividend?

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Common Core Math StandardsThe student is expected to:

A-APR.D.6

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), … using inspection, long division, … . Also A-APR.A.1, A-APR.B.3

Mathematical Practices

MP.2 Reasoning

Language ObjectiveWork in small groups to complete a compare and contrast chart for dividing polynomials.

COMMONCORE

COMMONCORE

HARDCOVER PAGES 231240

Turn to these pages to find this lesson in the hardcover student edition.

Dividing Polynomials

ENGAGE Essential Question: What are some ways to divide polynomials, and how do you know when the divisor is a factor of the dividend?Possible answer: You can divide polynomials using

long division or, for a divisor of the form x - a,

synthetic division. The divisor is a factor of the

dividend when the remainder is 0.

PREVIEW: LESSONPERFORMANCE TASKView the Engage section online. Discuss the photo and how the number of teams and the attendance are both functions of time. Then preview the Lesson Performance Task.

321

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Name

Class Date

Explore Evaluating a Polynomial Function

Using Synthetic Substitution

Polynomials can be written in something called nested form. A polynomial in nested form is written in such a way

that evaluating it involves an alternating sequence of additions and multiplications. For instance, the nested form of

p (x) = 4 x 3 + 3 x 2 + 2x + 1 is p (x) = x (x (4x + 3) + 2) + 1, which you evaluate by starting with 4, multiplying by

the value of x, adding 3, multiplying by x, adding 2, multiplying by x, and adding 1.

Given p (x) = 4 x3 + 3 x2 + 2x + 1, find p (-2) .

You can set up an array of numbers that captures the sequence of multiplications and

additions needed to find p (a). Using this array to find p (a) is called synthetic substitution.

Given p (x) = 4 x 3 + 3 x 2 + 2x + 1, find p (-2) by using synthetic substitution. The dashed

arrow indicates bringing down, the diagonal arrows represent multiplication by –2, and the

solid down arrows indicate adding.

The first two steps are to bring down the leading number, 4, then multiply by the value you

are evaluating at, -2.

Add 3 and –8.

Resource

Locker

A-APR.D.6 Rewrite simple rational expressions in different forms; write a(x)

___ b (x)

in the form q (x) + r (x) ________

b (x) ,

… using inspection, long division, … Also A-APR.A.1, A-APR.B.3

COMMONCORE

4

4

3

-8

21

-2

4

4

3

-8

21

-2

-5

Rewrite p (x) as p (x) = x (x (4x + 3) + 2) + 1.

Multiply. -2 · 4 = -8

Add. -8 + 3 = -5

Multiply. -5 · -2 = 10

Add. 10 + 2 = 12

Multiply. 12 · -2 = -24

Add. -24 + 1 = -23

Module 6

321

Lesson 5

6 . 5 Dividing Polynomials

Essential Question: What are some ways to divide polynomials, and how do you know when

the divisor is a factor of the dividend?

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321 Lesson 6 . 5

L E S S O N 6 . 5

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Multiply the previous answer by –2. Continue this sequence of steps until you reach the last addition.

p (-2) =

Reflect

1. Discussion After the final addition, what does this sum correspond to?

Explain 1 Dividing Polynomials Using Long DivisionRecall that arithmetic long division proceeds as follows.

Divisor 23 ← Quotient 12 ⟌ –––– 277 ← Dividend

24― 37

36― 1 ← Remainder

Notice that the long division leads to the result dividend _______ divisor = quotient + remainder ________ divisor . Using the

numbers from above, the arithmetic long division leads to 277 ___ 12 = 23 + 1 __ 12 . Multiplying through

by the divisor yields the result dividend = (divisor) (quotient) + remainder. (This can be used as

a means of checking your work.)

Example 1 Given a polynomial divisor and dividend, use long division to find the quotient and remainder. Write the result in the form dividend = (divisor) (quotient) + remainder and then carry out the multiplication and addition as a check.

(4 x 3 + 2 x 2 + 3x + 5) ÷ ( x 2 + 3x + 1) Begin by writing the dividend in standard form, including terms with a coefficient

of 0 (if any).

4 x 3 + 2 x 2 + 3x + 5

Write division in the same way as you would when dividing numbers.

x 2 + 3x + 1 ⟌ ––––––––––––––––– 4 x 3 + 2 x 2 + 3x + 5

4

4

3

-5

-8

2 1-2

104

4

3

-5

-8

2

10

1-2

12 -23

-24

-23

The final sum represents the value of p (x) where x = -2.




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Math BackgroundDivision of polynomials is related to division of whole numbers. Given

polynomials P (x) and D (x) , where D (x) ≠ 0, we can write P (x) _____ D (x) = Q (x) +

R (x) _____ D (x) , where the remainder R (x) is a polynomial whose degree is less than that of D (x) . (If the degree of R (x) were not less than the degree of D (x) , we would be able to continue dividing.) Equivalently, P (x) = Q (x) D (x) + R (x) . This last expression can be used to justify the Remainder Theorem. Notice that when D (x) is a linear divisor of the form x - a, the expression becomes P (x) = Q (x) (x - a) + r, where the remainder r is a real number.

EXPLORE Evaluating a Polynomial Function Using Synthetic Substitution

INTEGRATE TECHNOLOGYStudents have the option of completing the polynomial division activity either in the book

or online.

QUESTIONING STRATEGIESWhat operation are you doing and what are you finding when you use synthetic

substitution on the polynomial function p (x) to find p (a) ? You are dividing the polynomial function p (x)

by the quantity (x - a) , and you are finding the

value of p (a) .

EXPLAIN 1 Dividing Polynomials Using Long Division

AVOID COMMON ERRORSStudents may have difficulty relating the familiar long-division process for whole numbers to identifying the process for polynomials using the algorithm for finding dividend = (divisor)(quotient) + remainder. Point out that polynomial long division leads to this result: dividend ________ divisor = quotient + remainder _________ divisor , which is equivalent to dividend = (divisor)(quotient) + remainder if you multiply each term by the divisor. Showing an example of arithmetic long division alongside an example of polynomial division may help students make the connection.

PROFESSIONAL DEVELOPMENT

Dividing Polynomials 322


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Find the value you need to multiply the divisor by so that the first term matches with the first term of the dividend. In this case, in order to get 4 x 2 , we must multiply x 2 by 4x. This will be the first term of the quotient.

4x x 2 + 3x + 1 ⟌ ––––––––––––––––– 4 x 3 + 2 x 2 + 3x + 5

Next, multiply the divisor through by the term of the quotient you just found and subtract that value from the dividend. ( x 2 + 3x + 1) (4x) = 4 x 3 + 12 x 2 + 4x, so subtract 4 x 3 + 12 x 2 + 4x from 4 x 3 + 2 x 2 + 3x + 5.

4x x 2 + 3x + 1 ⟌ ––––––––––––––––– 4 x 3 + 2 x 2 + 3x + 5

――――――― - (4 x 3 + 12 x 2 + 4x)

-10 x 2 - x + 5

Taking this difference as the new dividend, continue in this fashion until the largest term of the remaining dividend is of lower degree than the divisor.

4x - 10

x 2 + 3x + 1 ⟌ ––––––––––––––––– 4 x 3 + 2 x 2 + 3x + 5

――――――― - (4 x 3 + 12 x 2 + 4x) -10 x 2 - x + 5

―――――――― - (-10 x 2 - 30x - 10) 29x + 15

Since 29x + 5 is of lower degree than x 2 + 3x + 1, stop. 29x + 15 is the remainder.

Write the final answer.

4 x 3 + 2 x 2 + 3x + 5 = ( x 2 + 3x + 1) (4x - 10) + 29x + 15

Check.

4x 3 + 2 x 2 + 3x + 5 = ( x 2 + 3x + 1) (4x - 10) + 29x + 15

= 4x 3 + 12 x 2 + 4x - 10 x 2 - 30x - 10 + 29x + 15

= 4 x 3 + 2 x 2 + 3x + 5

B (6 x 4 + 5 x 3 + 2x + 8) ÷ ( x 2 + 2x - 5) Write the dividend in standard form, including terms with a coefficient of 0.

Write the division in the same way as you would when dividing numbers.

x 2 + 2x - 5 ⟌ –––––––––––––––––––––– 6 x 4 + 5 x 3 + 0 x 2 + 2x + 8

6 x 4 + 5 x 3 + 0 x 2 + 2x + 8


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COLLABORATIVE LEARNING

Small Group ActivityHelp groups of students practice dividing polynomials using synthetic division. Provide each student with a different example of polynomial division, and ask them to show and explain the first step in dividing with synthetic division. Then have them pass the problem to another student, who writes the next step and explains it. They continue to pass the problem until each problem is completely solved and all steps are explained. The last student summarizes by giving the quotient and remainder in polynomial form. Encourage students to use these as examples of dividing polynomials when they write in their journals.

QUESTIONING STRATEGIESHow can you tell if you are finished solving a polynomial division problem? The remainder

has a degree less than the degree of the divisor, or

has degree 0.

What do you write as the final answer for a polynomial division problem? The answer

should be written as the product of factors plus the

remainder, where one factor is the divisor and the

other factor is the quotient.

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Divide.

6 x 2 - +

x 2 + 2x - 5 ⟌ –––––––––––––––––––––– 6 x 4 + 5 x 3 + 0 x 2 + 2x + 8

―――――――――― - (6 x 4 + 12 x 3 - 30 x 2 )

-7 x 3 + 30 x 2 + 2x

- (-7 x 3 )

――― + 8

- ( )

―――

Write the final answer.

6 x 4 + 5 x 3 + 2x + 8 =

Check.

Reflect

2. How do you include the terms with coefficients of 0?

Your Turn

Use long division to find the quotient and remainder. Write the result in the form dividend = (divisor) (quotient) + remainder and then carry out a check.

3. (15 x 3 + 8x - 12) ÷ (3 x 2 + 6x + 1)

7x

-14 x 2 + 35x

-121x + 228

44 x 2 + 88x - 220

( x 2 + 2x - 5) (6 x 2 - 7x + 44) - 121x + 228

6 x 4 + 5 x 3 + 2x + 8 = ( x 2 + 2x - 5) (6 x 2 - 7x + 44) -121x + 228

= 6 x 4 - 7 x 3 + 44 x 2 + 12 x 3 -14 x 2 + 88x - 30 x 2 + 35x - 220 - 121x + 228

= 6 x 4 + 5 x 3 + 2x + 8

44 x 2 - 33x

44

You represent the term with 0 as the coefficient, e.g, 0x.

5x - 10

3 x 2 + 6x + 1 ⟌ –––––––––––––––––––– 15 x 3 + 0 x 2 + 8x - 12

―――――――― - (15 x 3 + 30 x 2 + 5x)

-30 x 2 + 3x - 12

――――――――― - (-30 x 2 - 60x - 10)

63x - 2

(3 x 2 + 6x + 1) (5x - 10) + 63x - 2

= 15 x 3 - 30 x 2 + 30 x 2 - 60x + 5x - 10 +

63x - 2

= 15 x 3 + 8x - 12




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DIFFERENTIATE INSTRUCTION

Graphic OrganizersHave groups of students create graphic organizers to help them divide polynomials using synthetic division. Have them show how to organize a problem into a form similar to the one shown. Then have them use organizers to write each of the steps, explain what goes into each of the cells, and then interpret the results.



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4. (9 x 4 + x 3 + 11 x 2 - 4) ÷ ( x 2 + 16)

Explain 2 Dividing p (x) by x - a Using Synthetic DivisionCompare long division with synthetic substitution. There are two important things to notice. The first is that p (a) is equal to the remainder when p (x) is divided by x - a. The second is that the numbers to the left of p (a) in the bottom row of the synthetic substitution array give the coefficients of the quotient. For this reason, synthetic substitution is also called synthetic division.

Long Division Synthetic Substitution

3 x 2 + 10x + 20x - 2 ⟌ –––––––––––––––––– 3 x 3 + 4 x 2 + 0x + 10 ――――― - (3 x 3 - 6 x 2 ) 10 x 2 + 0x ―――――― - (10 x 2 - 20x) 20x + 10 ―――― -20x - 40 50

2 3 4 0 10

―――――― 6 20 40

3 10 20 50

Example 2 Given a polynomial p (x) , use synthetic division to divide by x - a and obtain the quotient and the (nonzero) remainder. Write the result in the form p (x) = (x - a) (quotient) + p (a) , then carry out the multiplication and addition as a check.

(7 x 3 - 6x + 9) ÷ (x + 5)

By inspection, a = -5. Write the coefficients and a in the synthetic division format.

Bring down the first coefficient. Then multiply and add for each column.

Write the result, using the non-remainder entries of the bottom row as the coefficients.

Check.

-5 7 0 -6 9 ――――――

-5 7 0 -6 9 ――――――――― -35 175 -845 7 -35 169 -836

(7 x 3 - 6x + 9) = (x + 5) (7 x 2 - 35x + 169) - 836

(7 x 3 - 6x + 9) = (x + 5) (7 x 2 - 35x + 169) - 836

= 7 x 3 - 35 x 2 - 35 x 2 - 175x + 169x + 845 - 836

= 7 x 3 - 6x + 9

9 x 2 + x - 133

x 2 + 16 ⟌ ––––––––––––––––––––––– 9 x 4 + x 3 + 11 x 2 + 0x - 4

―――――――― - (9 x 4 + 0 x 3 + 144 x 2 )

x 3 - 133 x 2 + 0x

――――――― - ( x 3 + 0 x 2 + 16x)

-133 x 2 - 16x - 4

――――――――― - (-133 x 2 + 0x - 2128)

-16x + 2124

( x 2 + 16) (9 x 2 + x - 133) - 16x + 2124

= 9 x 4 + x 3 - 133 x 2 + 144 x 2 + 16x - 2128 -

16x + 2124

= 9 x 4 + x 3 + 11 x 2 - 4



A2_MNLESE385894_U3M06L5 325 16/10/14 9:40 AM

2

2-2(2) -2(3)

7+(-4) 9+(-6)

3 3

7

-4 -6

9-2

bringdown 2

EXPLAIN 2 Dividing p (x) by x - a Using Synthetic Division

INTEGRATE MATHEMATICAL PRACTICESFocus on PatternsMP.8 Students should quickly see that there are patterns in synthetic division. Have students use arrows and expressions, if necessary, to help them understand the patterns. For example, (2 x 2 + 7x + 9) ÷ (x + 2) may be shown as:

Students should notice that the divisor is -2 because the form of the divisor is (x - a) , or (x - (-2) ) ; the rows are added; the remainder is the last digit in the last row, 3; and the last row includes the coefficients of the quotient, starting with the power of the variable decreased by 1. (2 x 2 + 7x + 9) ÷ (x + 2) = 2x + 3 + 3 _____ x + 2

LANGUAGE SUPPORT

Connect VocabularyHelp students understand how the method synthetic division is used as a symbolic representation of a polynomial division problem. Point out that the English word synthetic means not genuine, unnatural, artificial, or contrived. That implies they will have to understand how to interpret the numerical results in the last row of the synthetic division problem. To help students remember the value of a for the divisor (x - a) , show them how to form the equation x - a = 0, and then solve that equation for a. This procedure will remind students to use the correct sign for the divisor.

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B (4 x 4 - 3 x 2 + 7x + 2) ÷ (x - 1 _ 2 ) Find a. Then write the coefficients and a in the synthetic division format.

Find a =

4 0 -3 7 2 ―――――――

Bring down the first coefficient. Then multiply and add for each column.

4 0 -3 7 2 ――――――― 4

Write the result.

(4 x 4 - 3 x 2 + 7x + 2) =

Check.

Reflect

5. Can you use synthetic division to divide a polynomial by x 2 + 3? Explain.

Your Turn

Given a polynomial p (x) , use synthetic division to divide by x - a and obtain the quotient and the (nonzero) remainder. Write the result in the form p (x) = (x - a) (quotient) + p (a) . You may wish to perform a check.

6. (2 x 3 + 5 x 2 - x + 7) ÷ (x - 2)

7. (6 x 4 - 25x 3 - 3x + 5) ÷ (x + 1 _ 3 )

1 _ 2

2 1 -1 352 -2 6

(x - 1 _ 2

) (4 x 3 + 2 x 2 - 2x + 6) + 5

(4 x 4 - 3 x 2 + 7x + 2) = (x - 1 _ 2

) (4 x 3 + 2 x 2 - 2x + 6) + 5

= 4 x 4 + 2 x 3 - 2 x 2 + 6x - 2 x 3 - x 2 + x - 3 + 5

= 4 x 4 - 3 x 2 + 7x + 2

2 2 5 -1 7

―――――― 4 18 34

2 9 17 41

(x - 2) (2 x 2 + 9x + 17) + 41

= 2 x 3 - 4 x 2 + 9 x 2 - 18x + 17x - 34 + 41

= 2 x 3 + 5 x 2 - x + 7

No, the divisor must be a linear binomial in the form x - a; x 2 + 3 is a quadratic binomial.

1 _ 2

1 _ 2

- 1 _ 3

6 -25 0 -3 5

――――――― -2 9 -3 2

6 -27 9 -6 7

(x + 1 _ 3

) ( 6x 3 - 27x 2 + 9x - 6) + 7

= 6 x 4 + 2 x 3 - 27 x 3 - 9x 2 + 9x 2 + 3x - 6x

- 2 + 7

= 6x 4 - 25x 3 - 3x + 5




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QUESTIONING STRATEGIESHow can you recognize the quotient and remainder when using the synthetic division

method? The bottom row of the synthetic division

problem gives the coefficients of the quotient along

with the remainder of the division problem.



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Explain 3 Using the Remainder Theorem and Factor Theorem

When p (x) is divided by x - a, the result can be written in the form p (x) = (x - a) q (x) + r where q (x) is the quotient and r is a number. Substituting a for x in this equation gives p (a) = (a - a) q (a) + r. Since a - a = 0, this simplifies to p (a) = r. This is known as the Remainder Theorem.

If the remainder p (a) in p (x) = (x - a) q (x) + p (a) is 0, then p (x) = (x - a) q (x) , which tells you that x - a is a factor of p (x) . Conversely, if x - a is a factor of p (x) , then you can write p (x) as p (x) = (x - a) q (x) , and when you divide p (x) by x - a, you get the quotient q (x) with a remainder of 0. These facts are known as the Factor Theorem.

Example 3 Determine whether the given binomial is a factor of the polynomial p (x) . If so, find the remaining factors of p (x) .

p (x) = x 3 + 3 x 2 - 4x - 12; (x + 3)

Use synthetic division.

-3 1 3 -4 -12

―――――――― -3 0 12 1 0 -4 0

Since the remainder is 0, x + 3 is a factor.

Write q (x) and then factor it.

q (x) = x 2 - 4 = (x + 2) (x - 2)

So, p (x) = x 3 + 3 x 2 - 4x - 12 = (x + 2) (x - 2) (x + 3) .

p (x) = x 4 - 4 x 3 - 6 x 2 + 4x + 5; (x + 1)

Use synthetic division.

-1 1 -4 -6 4 5 ―――――――

1

Since the remainder is , (x + 1) a factor. Write q (x) .

q (x) =

Now factor q (x) by grouping.

q (x) =

=

=

=

So, p (x) = x 4 - 4 x 3 - 6 x 2 + 4x + 5 = .

-1 5 1-5

-5 -1 5

0 is

x 3 - 5 x 2 - x +5

x 3 - 5 x 2 - x +5

x 2 (x - 5) - (x - 5)

( x 2 - 1) (x - 5)

(x + 1) (x - 1) (x - 5)

(x + 1) (x + 1) (x - 1) (x - 5)

0



A2_MNLESE385894_U3M06L5 327 16/10/14 9:42 AM

EXPLAIN 3 Using the Remainder Theorem and Factor Theorem

QUESTIONING STRATEGIESHow are the Remainder Theorem and the Factor Theorem related? The Remainder

Theorem implies that if a polynomial p (x) is divided

by x - a, and the remainder p (a) = 0, then (x - a) is

a factor of the polynomial. The Factor Theorem says

that if (x - a) is a factor of p (x) , you can rewrite the

polynomial as the quotient q (x) times this factor, or

p (x) = (x - a) q (x) .

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Your Turn

Determine whether the given binomial is a factor of the polynomial p (x) . If it is, find the remaining factors of p (x) .

8. p (x) = 2 x 4 + 8 x 3 + 2x + 8; (x + 4)

9. p (x) = 3 x 3 - 2x + 5; (x - 1)

Elaborate

10. Compare long division and synthetic division of polynomials.

11. How does knowing one linear factor of a polynomial help find the other factors?

12. What conditions must be met in order to use synthetic division?

13. Essential Question Check-In How do you know when the divisor is a factor of the dividend?

-4 2 8 0 2 8

―――――――― -8 0 0 -8

2 0 0 2 0

Since the remainder is 0, (x + 4) is a factor.

q (x) = 2 x 3 + 2 = 2 ( x 3 + 1) = 2 (x + 1) ( x 2 - x + 1)

So, p (x) = 2 x 4 + 8 x 3 + 2x + 8

= 2 (x + 1) ( x 2 - x + 1) (x + 4)

Since the remainder is 6, x - 1 is not a factor.

1 3 0 -2 5

―――――― ――――― 3 3 1

3 1 6

3

The numbers generated by synthetic division are equal to the coefficients of the terms of

the polynomial quotient, including the remainder. They are essentially the same process.

If one linear factor of the polynomial is known, synthetic division can be used to find the

product of the other factors, which may be easily factorable.

The divisor must be a linear binomial with a leading coefficient of 1. The dividend must be

written in standard form with 0 representing any missing terms.

The divisor is a factor of the dividend when the remainder is 0.




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ELABORATE QUESTIONING STRATEGIES

When do you use synthetic substitution, and when do you use synthetic division? You use

synthetic substitution when you want to find the

function value of a polynomial for a certain number.

You use synthetic division when you want to find

the quotient of polynomial division.

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Have students work in pairs to complete a chart like the following, showing similarities and differences:

Alike Different

Long Division

Synthetic Substitution or Division

SUMMARIZE THE LESSONHow do you explain the process of synthetic division, and when and why it is useful?

What are possible sources of error in the process? Synthetic division is a division process that

uses only the coefficients of a polynomial. If the last

sum is 0, then the binomial is a factor of the

polynomial; possible errors are not bringing down

the first coefficient, forgetting to add instead of

subtract, and forgetting to include coefficients

that are 0.



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• Online Homework• Hints and Help• Extra Practice

Given p (x) , find p (-3) by using synthetic substitution.

1. p (x) = 8 x 3 + 7 x 2 + 2x + 4 2. p (x) = x 3 + 6 x 2 + 7x - 25

3. p (x) = 2 x 3 + 5 x 2 - 3x 4. p (x) = - x 4 + 5 x 3 - 8x + 45

Given a polynomial divisor and dividend, use long division to find the quotient and remainder. Write the result in the form dividend = (divisor)(quotient) + remainder. You may wish to carry out a check.

5. (18 x 3 - 3 x 2 + x -1) ÷ (x 2 - 4)

6. (6 x 4 + x 3 - 9x + 13) ÷ (x 2 + 8)

Evaluate: Homework and Practice

-3 8 7 2 4

―――――――― -24 51 -159

8 -17 53 -155

p (-3) = -155

-3 2 5 -3 0

―――――――― -6 3 0

2 -1 0 0

p (-3) = 0

-3 -1 5 0 -8 45

―――――――――――― 3 -24 72 -192

-1 8 -24 64 -147

p (-3) = -147

-3 1 6 7 -25

――――――― -3 -9 6

1 3 -2 -19

p (-3) = -19

18x - 3

x 2 - 4 ⟌ ––––––––––––––––––– 18 x 3 - 3 x 2 + x - 1

――――――――― - (18 x 3 + 0 x 2 - 72x)

-3 x 2 + 73x - 1

―――――――― - (-3 x 2 + 0x + 12)

73x - 13

Check.

( x 2 - 4) (18x - 3) + 73x - 13

= 18 x 3 - 72x - 3 x 2 + 12 + 73x - 13

= 18 x 3 - 3 x 2 + x - 1

6 x 2 + x - 48

x 2 + 8 ⟌ ––––––––––––––––––––––––– 6 x 4 + x 3 + 0 x 2 - 9x + 13

―――――――――――― - (6 x 4 + 0 x 3 + 48 x 2 )

x 3 - 48 x 2 - 9x

――――――――― - ( x 3 + 0 x 2 + 8x)

-48 x 2 - 17x + 13

――――――――― - (-48 x 2 + 0x - 384)

-17x + 397

Check.

( x 2 + 8) (6 x 2 + x - 48) - 17x + 397

= 6 x 4 + x 3 - 48 x 2 + 48 x 2 + 8x - 384 - 17x + 397

= 6 x 4 + x 3 - 9x + 13


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A2_MNLESE385894_U3M06L5.indd 329 3/19/14 11:57 AMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

1–15 1 Recall of Information MP.2 Reasoning

16–19 2 Skills/Concepts MP.4 Modeling

20 2 Skills/Concepts MP.2 Reasoning

21 3 Strategic Thinking MP.2 Reasoning

22 3 Strategic Thinking MP.6 Precision

23 3 Strategic Thinking MP.2 Reasoning

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreEvaluating a Polynomial Function Using Synthetic Substitution

Exercises 1–4

Example 1Dividing Polynomials Using Long Division

Exercises 5–8

Example 2Dividing p (x) by x - a Using Synthetic Division

Exercises 9–11

Example 3Using the Remainder Theorem and Factor Theorem

Exercises 12–15

AVOID COMMON ERRORSStudents might make errors in signs when doing synthetic division and synthetic substitution because values are added rather than subtracted as in long division. Remind them that terms are always added for synthetic substitution and synthetic division.

AVOID COMMON ERRORSStudents may be confused about when to use synthetic division and when to use long division. Point out that for a divisor other than a linear binomial with leading coefficient 1, long division is the best method. It may, however, be possible to divide the dividend and divisor by a constant to make the leading coefficient 1.

329 Lesson 6 . 5


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7. ( x 4 + 6x - 2.5) ÷ ( x 2 + 3x + 0.5)

8. ( x 3 + 250 x 2 + 100x) ÷ ( 1 _ 2 x 2 + 25x + 9)

Given a polynomial p (x) , use synthetic division to divide by x - a and obtain the quotient and the (nonzero) remainder. Write the result in the form p (x) = (x - a) (quotient) + p (a) . You may wish to carry out a check.

9. (7 x 3 - 4 x 2 - 400x - 100) ÷ (x - 8)

10. (8 x 4 - 28.5 x 2 - 9x + 10) ÷ (x + 0.25)

11. (2.5 x 3 + 6 x 2 - 5.5x - 10) ÷ (x + 1)

x 2 - 3x + 8.5

x 2 + 3x + 0.5 ⟌ –––––––––––––––––––––––– x 4 + 0 x 3 + 0 x 2 + 6x - 2.5

――――――――――― - ( x 4 + 3 x 3 + 0.5x 2 )

-3 x 3 - 0.5 x 2 + 6x

――――――――― - (-3 x 3 - 9 x 2 - 1.5x)

8.5 x 2 + 7.5x - 2.5

――――――――― - (8.5 x 2 + 25.5x + 4.25)

-18x - 6.75

Check.

( x 2 + 3x + 0.5) ( x 2 - 3x + 8.5) - 18x - 6.75

= x 4 - 3 x 3 + 8.5 x 2 + 3 x 3 - 9 x 2 + 25.5x + 0.5 x 2

- 1.5x + 4.25 - 18x - 6.75

= x 4 + 6x - 2.5

2x + 400

1 _ 2

x 2 + 25x + 9 ⟌ ––––––––––––––––––––– x 3 + 250 x 2 + 100x + 0

―――――――― ――――――― - ( x 3 + 50 x 2 + 18x)

200 x 2 + 82x + 0

――――――――――― - (200 x 2 + 10, 000x + 3600)

-9918x - 3600

Check.

( 1 _ 2

x 2 + 25x + 9) (2x + 400) - 9918x - 3600

= x 3 + 200 x 2 + 50 x 2 + 10, 000x + 18x + 3600

- 9918x - 3600

= x 3 + 250 x 2 + 100x

8 7 -4 -400 -100

――――――――― 56 416 128

7 52 16 28

(x - 8) (7 x 2 + 52x + 16) + 28

= 7 x 3 - 56 x 2 + 52 x 2 - 416x + 16x - 128 + 28

= 7 x 3 - 4 x 2 - 400x - 100

(x + 0.25) (8 x 3 - 2 x 2 - 28x - 2) + 9.5

= 8 x 4 + 2 x 3 - 2 x 3 - 0.5 x 2 - 28 x 2 - 7x - 2x

- 0.5 + 9.5

= 8 x 4 - 28.5 x 2 - 9x + 10

(x + 1) (2.5 x 2 + 3.5x - 9) - 1

= 2.5 x 3 + 2.5 x 2 + 3.5 x 2 + 3.5x - 9x - 9 - 1

= 2.5 x 3 + 6 x 2 - 5.5x - 10

-0.25 8 0 -28.5 -9 10

――――――――――――― -2 0.5 7 -0.5

8 -2 -28 -2 9.5

-1 2.5 6 -5.5 -10

――――――――― -2.5 -3.5 9

2.5 3.5 -9 -1




A2_MNLESE385894_U3M06L5 330 6/27/14 7:01 PM

AVOID COMMON ERRORSA common error when doing synthetic division is to subtract the second row rather than adding it. Show students a long division problem alongside its solution, using synthetic division to emphasize that the results will be different if they make this error.



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Determine whether the given binomial is a factor of the polynomial p (x) . If so, find the remaining factors of p (x) .

12. p (x) = x 3 + 2 x 2 - x - 2; (x + 2) 13. p (x) = 2 x 4 + 6 x 3 - 5x - 10; (x + 2)

14. p (x) = x 3 - 22 x 2 + 157x - 360; (x - 8)

15. p (x) = 4 x 3 - 12 x 2 + 2x - 5; (x - 3)

16. The volume of a rectangular prism whose dimensions are binomials with integer coefficients is modeled by the function V (x) = x 3 - 8 x 2 + 19x - 12 . Given that x - 1 and x - 3 are two of the dimensions, find the missing dimension of the prism.

-2 1 2 -1 -2

―――――― -2 0 2

1 0 -1 0

x + 2 is a factor.

x 2 - 1 = (x + 1) (x - 1)

So, p (x) = x 3 + 2 x 2 - x - 2

= (x + 1) (x - 1) (x + 2) .

-2 2 6 0 -5 -10

―――――――――― -4 -4 8 -6

2 2 -4 3 -16

x + 2 is not a factor.

1 1 -8 19 -12

―――――――― 1 -7 12

1 -7 12 0

This gives the expression x 2 -7x + 12.

3 1 -7 12

――――― 3 -12

1 -4 0

This gives the expression x - 4, which is the missing dimension.

3 4 -12 2 -5

―――――――― 12 0 6

4 0 2 1

x - 3 is not a factor.

8 1 -22 157 -360

―――――――――― 8 -112 360

1 -14 45 0

x - 8 is a factor.

x 2 - 14x + 45 = (x - 5) (x - 9)

So, p (x) = x 3 - 22 x 2 + 157x - 360 = (x - 5) (x - 9) (x - 8) .


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331 Lesson 6 . 5


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17. Given that the height of a rectangular prism is x + 2 and the volume is x 3 - x 2 - 6x, write an expression that represents the area of the base of the prism.

18. Physics A Van de Graaff generator is a machine that produces very high voltages by using small, safe levels of electric current. One machine has a current that can be modeled by l (t) = t + 2 , where t > 0 represents time in seconds. The power of the system can be modeled by P (t) = 0.5 t 3 + 6 t 2 + 10t. Write an expression that represents the voltage of the system. Recall that V = P __ l .

19. Geometry The volume of a hexagonal pyramid is modeled by the function V (x) = 1 __ 3 x 3 + 4 __ 3 x 2 + 2 __ 3 x - 1 __ 3 . Given the height x + 1, use polynomial division to find an expression for the area of the base.

(Hint: For a pyramid, V = 1 __ 3 Bh.)

20. Explain the Error Two students used synthetic division to divide 3 x 3 - 2x - 8 by x - 2. Determine which solution is correct. Find the error in the other solution.

A. B.

2 3 0 -2 -8 6 12 20 3 6 10 12

-2 3 0 -2 -8

――――――― -6 12 -20

3 -6 10 -28

-2 1 -1 -6 0

―――――――― -2 6 0

1 -3 0 0

So, the area can be represented by x 2 - 3x.

-2 0.5 6 10 0 ――――――― -1 -10 0

0.5 5 0 0

The voltage can be represented by 0.5 t 2 + 5t.

V (x) = 1 _ 3 ( x 3 + 4 x 2 + 2x - 1) .

-1 1 4 2 -1 ―――――― -1 -3 1

1 3 -1 0

So, the area of the base can be represented by x 2 + 3x - 1 .

Student A is correct. Student B used the incorrect sign of a.


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A2_MNLESE385894_U3M06L5 332 6/9/15 12:07 AM

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Emphasize the conditions that must be met to use synthetic division: The divisor must be a linear binomial with a leading coefficient of 1. The dividend must be written in standard form with 0 representing any missing terms.

Discuss the characteristics of synthetic division that make it synthetic: there are no variables, only coefficients; and addition is used instead of subtraction.

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Point out that students should review how to do synthetic division and long division when the dividend is missing terms for some powers of the variable. Emphasize that they must include the missing terms written as zero times the power of the variable to complete the division.



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H.O.T. Focus on Higher Order Thinking

21. Multi-Step Use synthetic division to divide p (x) = 3 x 3 - 11 x 2 - 56x - 50 by (3x + 4) . Then check the solution.

22. Critical Thinking The polynomial a x 3 + b x 2 + cx + d is factored as 3 (x - 2) (x + 3) (x - 4) . What are the values of a and d? Explain.

23. Analyze Relationships Investigate whether the set of whole numbers, the set of integers, and the set of rational numbers are closed under each of the four basic operations. Then consider whether the set of polynomials in one variable is closed under the four basic operations, and determine whether polynomials are like whole numbers, integers, or rational numbers with respect to closure. Use the table to organize.

Whole Numbers Integers

Rational Numbers Polynomials

Addition

Subtraction

Multiplication

Division (by nonzero)

Rewrite 3x + 4 as 3 (x + 4 _ 3

) .

- 4 _

3 3 -11 -56 -50

――――――――― -4 20 48

3 -15 -36 -2

The quotient needs to be divided by 3.

3 x 2 - 15x - 36 __ 3

= x 2 - 5x - 12

Check.

(3x + 4) ( 3 x 2 - 15x - 36 __ 3

) - 2

= (3x + 4) ( x 2 - 5x - 12) - 2

= 3 x 3 + 4 x 2 - 15 x 2 - 20x - 36x - 48 - 2

= 3 x 3 - 11 x 2 - 56x - 50

a = 3; d = 72; The value of a is the product of the GCF and the coefficients of the x-terms of each factor. The value of d is the product of the GCF and the constant terms of each factor.

Yes

No

Yes

No YesNo No

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Polynomials are similar to integers with respect to closure. They are closed under each operation except division.




PEER-TO-PEER DISCUSSIONInstruct one student in each pair to solve a polynomial division problem using long division, while the other student solves it by using synthetic division. Then have students switch roles and repeat the exercise for a new division problem with a polynomial that does not have a linear factor. Then have them discuss their results and any preferences they have for one method or the other.

JOURNALHave students describe a mnemonic device that can help them remember the steps in synthetic division.

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Lesson Performance TaskThe table gives the attendance data for all divisions of NCAA Women’s Basketball.

NCAA Women’s Basketball Attendance

SeasonYears since 2006–2007

Number of teams in all 3 divisions

Attendance (in thousands) for

all 3 divisions

2006–2007 0 1003 10,878.3

2007–2008 1 1013 11,120.8

2008–2009 2 1032 11,160.3

2009–2010 3 1037 11,134.7

2010–2011 4 1048 11,160.0

2011–2012 5 1055 11,201.8

Enter the data from the second, third, and fourth columns of the table and perform linear regression on the data pairs (t, T) and cubic regression on the data pairs (t, A) where t = years since the 2006–2007 season, T = number of teams, and A = attendance (in thousands).

Then create a model for the average attendance per team: A avg (t) = A (t) ___

T (t) . Carry out the division to write A avg (t) in the form quadratic quotient + remainder _______

T (t) .

Use an online computer algebra system to carry out the division of A (t) by T (t) .

Models:

T (t) = 10.57t + 1005

A (t) = 13.80 t 3 - 121.6 t 2 + 329.8t + 10,880

Online computer algebra system result:

A avg (t) = 1.30558 t 2 - 135.64t + 12,927.9 - 12,981,600

__ 10.57t + 1005




AVOID COMMON ERRORSStudents may try to include the “3 divisions” in their calculation, perhaps by dividing the number of teams or attendance by 3. Explain to students that a division is a group of schools that compete against each other and that there are 3 such groups, or divisions, for college basketball. Explain that this number is irrelevant for this calculation. What is important is that the numbers in the table represent the total number of teams and the total attendance for all of women’s collegiate basketball.

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Have students highlight the remainder in the final function Aavg(t). Ask them if the remainder is zero or nonzero and what a nonzero remainder means. Have students discuss the significance of a remainder for a function that describes attendance per team. If the attendance is 315.8 fans per team, ask students if the 0.8 represents an actual person or if it results from a limitation of the model.

Scoring Rubric

2 points: Student correctly solves the problem and explains his/her reasoning.

1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.

0 points: Student does not demonstrate understanding of the problem.

EXTENSION ACTIVITY

Have students research the attendance for a specific NCAA women’s basketball team for a single season. Have them compare that numb er to the value calculated from the model Aavg(t) for the same year. Ask the students how accurate the model is in predicting the attendance for that team and what some of the sources of error might be



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Dividing Polynomials