Dividing Polynomials - Denton ISD [PDF]

LESSON

7.4

Name

Dividing Polynomials

Class

7.4

A2.7.C Determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two. Also A2.7.D

The student is expected to: A2.7.C

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

Mathematical Processes

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

A2.1.F



Analyze mathematical relationships to connect and communicate mathematical ideas.

Language Objective

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

Rewrite p(x) as p(x) = x(x(4x + 3) + 2) + 1. -2 · 4 = -8

Multiply.

-8 + 3 = -5

Add.

2.D.1, 2.I.4, 3.E, 3.H.3

Multiply. -5 · (-2) = 10

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

10 + 2 = 12

Add.

ENGAGE

Multiply.

12 · (-2) = -24

Add.

-24 + 1 = -23

© Houghton Mifflin Harcourt Publishing Company

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) = 4x 3 + 3x 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. -2

4

3

2

1

2

1

-8 4



Add 3 and –8. -2

PREVIEW: LESSON PERFORMANCE TASK View 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.

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p(-2). 2 2x + 1, find + 3x + ) + 2 + 1. x(4x + 3 (x) = x p(x) as p Rewrite -8 -2 · 4 = Multiply. -5 -8 + 3 = Add. = 10 -5 · -2 Multiply. 12 10 + 2 = Add. -24 12 · 2 = and Multiply. lications = -23 ce of multip etic substitution. -24 + 1 es the sequen synth Add. that captur find p(a) is called d to of numbers . The dashe up an array p(a). Using this array tic substitutionby –2, and the You can set find to n using synthe needed multiplicatio p(-2) by additions represent + 1, find 2 3 + 3x + 2x al arrows x 4 = the diagon ) x ( you Given p ng down, by the value tes bringi multiply adding. arrow indica er, 4, then s indicate numb arrow g leadin solid down down the are to bring 1 two steps 2 The first 3 ting at, -2. 4 are evalua -2 3

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

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

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2

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4

1

-8 4

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Lesson 4

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Lesson 7.4

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Resource Locker

Evaluating a Polynomial Function Using Synthetic Substitution

Explore

Determine the quotient of a polynomial of degree three and of degree four when divided by a polynomial of degree one and of degree two. Also A2.7.D

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.

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?

Texas Math Standards

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

Date

22/02/14

10:30 AM

1/11/15 10:50 AM



Multiply the previous answer by –2. -2

4

3 -8

Continue this sequence of steps until you reach the last addition.

1

-2

10

-5

4



2



4

4

3

2

1

-8

10

-24

-5

12

-23

EXPLORE Evaluating a Polynomial Function Using Synthetic Substitution

p(-2) = -23

INTEGRATE TECHNOLOGY Students have the option of completing the polynomial division activity either in the book or online.

Reflect

1.

Discussion After the final addition, what does this sum correspond to? The final sum represents the value of p(x) where x = -2.

QUESTIONING STRATEGIES

Dividing Polynomials Using Long Division

Explain 1

What 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).

Recall that arithmetic long division proceeds as follows.

Divisor

––––

23 ← Quotient 12 ⟌ 277 ← Dividend 24

― 37 36

―

1 ← Remainder

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

divisor

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

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

Example 1



EXPLAIN 1

© Houghton Mifflin Harcourt Publishing Company

a means of checking your work.)

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

(4x 3 + 2x 2 + 3x + 5) ÷ (x 2 + 3x + 1) Begin by writing the dividend in standard form, including terms with a coefficient of 0 (if any). 4x 3 + 2x 2 + 3x + 5

Dividing Polynomials Using Long Division AVOID COMMON ERRORS

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

–––––––––––––––––

x 2 + 3x + 1 ⟌ 4x 3 + 2x 2 + 3x + 5

Module 7

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Lesson 4

PROFESSIONAL DEVELOPMENT A2_MTXESE353930_U3M07L4.indd 392

Math Background

Division of polynomials is related to division of whole numbers. Given P(x) R(x) polynomials P(x) and D(x), where D(x) ≠ 0, we can write _____ = Q(x) + _____ , D(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.

2/20/14 9:32 PM

Students 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 remainder , dividend = quotient + _________ leads to this result: ________ divisor 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.

Dividing Polynomials 392

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 4x 2, we must multiply x 2 by 4x. This will be the first term of the quotient. 4x

QUESTIONING STRATEGIES

–––––––––––––––––

How 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.

x 2 + 3x + 1 ⟌ 4x 3 + 2x 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) = 4x 3 + 12x 2 + 4x, so subtract 4x 3 + 12x 2 + 4x from 4x 3 + 2x 2 + 3x + 5.

––––––––––––––––––

4x x 2 + 3x + 1 ⟌ 4x 3 + 2x 2 + 3x + 5 -4___ x 3 + 12x 2 + 4x -10x 2 - x + 5

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.

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

――――――――― -10x - x + 5 2

-(-10x - 30x - 10)

―――――――― 29x + 15 2

Since 29x + 5 is of lower degree than x 2 + 3x + 1, stop. 29x + 15 is the remainder. Write the final answer. 4x 3 + 2x 2 + 3x + 5 = (x 2 + 3x + 1)(4x - 10) + 29x + 15 Check. 4x 3 + 2x 2 + 3x + 5 = (x 2 + 3x + 1)(4x - 10) + 29x + 15

© Houghton Mifflin Harcourt Publishing Company

= 4x 3 + 12x 2 + 4x - 10x 2 - 30x - 10 + 29x + 15 = 4x 3 + 2x 2 + 3x + 5

B

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

6x 4 + 5x 3 + 0x 2 + 2x + 8 Write the division in the same way as you would when dividing numbers.

––––––––––––––––––––––

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

Module 7

393

Lesson 4

COLLABORATIVE LEARNING A2_MTXESE353930_U3M07L4.indd 393

Small Group Activity Help 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.

393

Lesson 7.4

1/13/15 5:32 AM

Divide. 6x 2 - 7x + 44

–––––––––––––––––––––––––––– +8

x 2 + 2x - 5 ⟌ 6x 4 + 5x 3 + 0x 2 + 2x -(6x 4 + 12x 3 - 30x 2)

――――――――――― -7x 3 + 30x 2 + 2x

(

2 - -7x 3 -14x + 35x

)

―――― 44x 2 - 33x + 8

-

( 44x + 88x - 220 ) 2

――― -121x + 228

Write the final answer. 6x 4 + 5x 3 + 2x + 8 = (x 2 + 2x - 5)(6x 2 - 7x + 44) - 121x + 228 Check.

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

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

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

2.

How do you include the terms with coefficients of 0? You represent the term with 0 as the coefficient, e.g, 0x.

3.

(15x 3 + 8x - 12) ÷ (3x 2 + 6x + 1)

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

5x - 10

––––––––––––––––––––––

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

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

-(15x 3 + 30x 2 + 5x)

63x - 2

――――――――――

= 15x 3 + 8x - 12

-30x + 3x - 12 2

-(-30x 2 - 60x - 10)

© Houghton Mifflin Harcourt Publishing Company

Your Turn

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

――――――――― 63x - 2

Module 7

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Lesson 4

DIFFERENTIATE INSTRUCTION A2_MTXESE353930_U3M07L4.indd 394

Graphic Organizers

1/13/15 5:32 AM

Have 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.

Dividing Polynomials 394

(9x 4 + x 3 + 11x 2 - 4) ÷ (x 2 + 16)

4.

EXPLAIN 2

x2

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

4

9

-4

-6

7+(-4)

2

-2(2)

3

-2(3)

3

2

)

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

2

-(x 3 +

0x 2 + 16x)

――――――――

-133x 2 - 16x -

4

-(-133x + 0x - 2128)

―――――――――― 2

-16x + 2124

Explain 2

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

Compare 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

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

―――――

2

3 4 0 10 6 20 40 ―――――― 3 10 20 50

――――――

9+(-6) 3

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. 3 (2x 2 + 7x + 9) ÷ (x + 2) = 2x + 3 + _____ x+2

16x + 2124

x - 133x + 0x 3

Example 2

© Houghton Mifflin Harcourt Publishing Company

bring down 2

7

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

2

-(9x + 0x + 144x

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, (2x 2 + 7x + 9) ÷ (x + 2) may be shown as: 2

3

―――――――――――― 4

INTEGRATE MATHEMATICAL PROCESSES Focus on Patterns

-2

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

9x 2 + x - 133

+ 16 ⟌––––––––––––––––––––––––––––– 9x +   x + 11x +   0x 4



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.

(7x 3 - 6x + 9) ÷ (x + 5) By inspection, a = -5. Write the coefficients and a in the synthetic division format.

-5

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

-5

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

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

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

(7x 3 - 6x + 9) = (x + 5)(7x 2 - 35x + 169) - 836 = 7x 3 - 35x 2 - 35x 2 - 175x + 169x + 845 - 836 = 7x 3 - 6x + 9

Module 7

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Lesson 4

LANGUAGE SUPPORT A2_MTXESE353930_U3M07L4.indd 395

Connect Vocabulary

Help 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.

395

Lesson 7.4

1/13/15 5:32 AM

B

(4x 4 - 3x 2 + 7x + 2) ÷

(x - _21 )

QUESTIONING STRATEGIES

Find a. Then write the coefficients and a in the synthetic division format.

How 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.

1 Find a = _ 2 1 _ 2 4 0 -3 7 2

―――――――

Bring down the first coefficient. Then multiply and add for each column. 1 _ 2 4 0 -3 7 2 2 1 -1 3 ――――――― 4 2 -2 6 5 Write the result.

(

)

1 (4x 3 + 2x 2 - 2x + 6) + 5 (4x 4 - 3x 2 + 7x + 2)= x - _ 2 Check.

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

(x - _12 )(4x + 2x - 2x + 6) + 5 3

2

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

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

5.

Can you use synthetic division to divide a polynomial by x 2 + 3? Explain. No, the divisor must be a linear binomial in the form x - a; x 2 + 3 is a quadratic binomial.

Your Turn

6.

(2x 3 + 5x 2 - x + 7) ÷ (x - 2) 2

2

5 -1

4

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

7

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

18 34

―――――― 2 9 17 41 7.

(6x 4 - 25x 3 - 3x + 5) ÷ 1 -_ 3

6 -25 0 -3

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

(x + _31 )

(x + _13 )(6x - 27x + 9x - 6) + 7 3

5

2

© Houghton Mifflin Harcourt Publishing Company

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.

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

-2 9 -3 2 ――――――― 6 -27 9 -6 7

4

3

3

-2+7 = 6x 4 - 25x 3 - 3x + 5

Module 7

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396

Lesson 4

1/11/15 12:25 PM

Dividing Polynomials 396

Using the Remainder Theorem and Factor Theorem

Explain 3

EXPLAIN 3

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.

Using the Remainder Theorem and Factor 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.

QUESTIONING STRATEGIES How 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).

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

Example 3



p(x) = x 3 + 3x 2 - 4x - 12; (x + 3) Use synthetic division. -3

3 -4 -12 -3 0 12 ―――――――― 1 0 -4 0 1

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



p(x) = x 4 - 4x 3 - 6x 2 + 4x + 5; (x + 1) Use synthetic division.

© Houghton Mifflin Harcourt Publishing Company

-1

-4 -6 4 5 -1 5 1-5 ――――――― -5 -1 5 0 1 1

Since the remainder is

0 , (x + 1) is a factor. Write q(x).

q(x) = x - 5x - x +5 3

2

Now factor q(x) by grouping. q(x) = x 3 - 5x 2 - x +5 = x 2(x - 5) - (x - 5) =

(x 2 - 1)(x - 5)

=

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

So, p(x) = x 4 - 4x 3 - 6x 2 + 4x + 5 = (x + 1)(x + 1)(x - 1)(x - 5) .

Module 7

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Your Turn

EXPLAIN 4

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

-4

9.

2

8 0 2 8 -8 0 0 -8 ―――――――― 2 0 0 2 0

0 -2 5 3 3 1 ―――――― ――――― 3 3 1 6 3

Explain 4 Example 3



q(x) = 2x 3 + 2 = 2(x 3 + 1) =2(x + 1)(x 2 - x + 1) So, p(x) = 2x 4 + 8x 3 + 2x + 8

INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections

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

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

1

Solving a Real-World Problem Using Polynomial Division

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

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

Discuss with students how synthetic division produces a “reduced polynomial,” or a factorization of the polynomial that includes another polynomial with lesser degree. In turn, that reduced polynomial can itself be further factored until it is factored completely. This gives all the zeros of the associated polynomial function. In the case of a real-world polynomial, factoring completely may include estimated zeros.

Solving a Real-World Problem Using Polynomial Division

Solve the problem using polynomial division.

The profit P (in millions of dollars) for a clock factory can be modeled by P = -13x 3 + 21x where x is the number of clocks produced (in millions). The company now produces 1 million clocks and makes a profit of $8,000,000, but would like to cut back on production. What lesser number of clocks could the company produce and still make the same profit? 8 = -13x 3 + 21x 0 = 13x 3 - 21x + 8

1

0 -21 8 13 13 -8 ―――――― 13 13 -8 0

13

So, (x - 1)(13x 2 + 13x - 8) = 0.

__

± √ b 2 - 4ac ____________ Recall that the quadratic formula is x = -b . 2a

Use the quadratic formula to find that x ≈ 0.4 is the other positive solution. The company could still make the same profit producing about 400,000 clocks.

Module 7

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398

© Houghton Mifflin Harcourt Publishing Company

You know that x = 1 is a solution to the equation. This implies that x - 1 is a factor of 13x 3 - 21x + 8. Use synthetic division to find the other factors.

Lesson 4

1/13/15 5:32 AM

Dividing Polynomials 398

B

QUESTIONING STRATEGIES How is synthetic division used to solve a real-world polynomial equation? Synthetic division helps you quickly find the factors of the polynomial (the remainders are 0 at each step of the division), which in turn give all the solutions to the polynomial equation using the zero-product property.

The profit P (in thousands of dollars) for a jewelry store can be modeled by P = -16x 3 + 32x where x is the number of pieces of jewelry produced (in thousands). The company now produces 1000 pieces and makes a profit of $16,000, but would like to cut back on production. What lesser number of pieces of jewelry could the store produce and still make the same profit? 16 = -16x 3 + 32x 0 = 16x 3 - 32x + 16 You know that x = 1 is a solution to the equation. This implies that x - 1 is a factor of

16x 3 - 32x + 16 . Use synthetic division to find the other factors. 0 -32 16 16 16 -16 ―――――― 16 16 -16 0

1

16

So, (x - 1) 16x 2 + 16x - 16

= 0.

Factor out the GCF. 0 = (x - 1) 16x 2 + 16x - 16

16(x - 1)(x 2 + x - 1)

=

__

± √ b 2 - 4ac ____________ Recall that the quadratic formula is x = -b . 2a

Use the quadratic formula to find that x ≈ 0.6 is the other positive solution. The company could still make the same profit producing about 600 pieces of jewelry.

© Houghton Mifflin Harcourt Publishing Company

Your Turn

10. The total number of dollars donated each year to a small charitable organization has followed the trend d(t) = 2t 3 + 10t 2 + 2000t + 10,000, where d is dollars and t is the number of years since 1990. The average number of dollars per donor can be expressed by t + 5. Write an expression describing the total number of donors each year.

-5

2000 10,000

-10

0 -10, 000

The expression is 2t 2 + 2000.

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―――――――――― 2 0 2000 0

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Elaborate

ELABORATE

11. Compare long division and synthetic division of polynomials. 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.

QUESTIONING STRATEGIES

12. How does knowing one linear factor of a polynomial help find the other factors? If one linear factor of the polynomial is known, synthetic division can be used to find the

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.

product of the other factors, which may be easily factorable. 13. What conditions must be met in order 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. 14. Essential Question Check-In How do you know when the divisor is a factor of the dividend? The divisor is a factor of the dividend when the remainder is 0.

INTEGRATE MATHEMATICAL PROCESSES Focus on Communication Have students work in pairs to complete a chart like the following, showing similarities and differences:

Evaluate: Homework and Practice • Online Homework • Hints and Help • Extra Practice

Given p(x), find p(-3) by using synthetic substitution. 1.

p(x) = 8x + 7x + 2x + 4 3

-3

8

2

7

2

2.

4

51 -159

-17

53 -155

p(x) = x 3 + 6x 2 + 7x - 25

Long Division

-3

Synthetic Substitution or Division

―――――――― 8

p(-3) = -155 3.

p(x) = 2x 3 + 5x 2 - 3x

-3

2

-3

1

-6

3

0

-1

0

0

p(-3) = 0

7

-3

-9

-25

3

6

-2 -19

p(-3) = -19

0

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

-3

Module 7

-1

5

0

-8

45

3 -24 72 -192 ―――――――――――― -1 8 -24 64 -147

―――――――― 2

6

―――――――

4.

5

1

p(-3) = -147

Exercise

Depth of Knowledge (D.O.K.)

Mathematical Processes

1–17

1 Recall of Information

1.F Analyze relationships

18–21

2 Skills/Concepts

1.A Everyday life

22

2 Skills/Concepts

1.F Analyze relationships

23

3 Strategic Thinking

1.F Analyze relationships

24

3 Strategic Thinking

1.D Multiple representations

25

3 Strategic Thinking

1.F Analyze relationships

Different

SUMMARIZE THE LESSON How 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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Alike

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Dividing Polynomials 400

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.

EVALUATE

5.

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

x

2

Check.

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

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

2

-(18x 3 + 0x 2 - 72x)

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

――――――――― -3x 2 + 73x - 1

-(-3x + 0x + 12)

――――――――

ASSIGNMENT GUIDE Concepts and Skills

Practice

Explore Evaluating a Polynomial Function Using Synthetic Substitution

Exercises 1–4

Example 1 Dividing Polynomials Using Long Division

Exercises 5–8

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

Exercises 9–12

Example 3 Using the Remainder Theorem and Factor Theorem

Exercises 13–17

73x - 13

6.

(6x

4

+ x - 9x + 13) ÷ (x 2 + 8) 6x 2 + x - 48 3

Check.

–––––––––––––––––––––––––

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

x 2 + 8 ⟌ 6x 4 + x 3 + 0x 2 - 9x + 13 -(6x 4 + 0x 3 + 48x 2)

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

――――――――――――

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

x 3 - 48x 2 - 9x

-(x 3 + 0x 2 + 8x)

――――――――― -48x 2 - 17x + 13

-(-48x 2 + 0x - 384)

――――――――― 7.

AVOID COMMON ERRORS Students 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.

(x x

Exercises 18–21

-17x + 397

+ 6x - 2.5) ÷ (x + 3x + 0.5)

4

2

2

x 2 - 3x + 8.5

Check.

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

3

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

2

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

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Example 4 Solving a Real-World Problem Using Polynomial Division

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

2

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

――――――――――― -3x 3 - 0.5x 2 + 6x

- 1.5x + 4.25 - 18x - 6.75

-(-3x 3 - 9x 2 - 1.5x)

= x 4 + 6x - 2.5

―――――――――

8.5x 2 + 7.5x - 2.5

-(8.5x 2 + 25.5x + 4.25)

―――――――――

8.

-18x - 6.75 1 x 2 + 25x + 9 _ 2 2x + 400

(x 3 + 250x 2 + 100x) ÷

_1 x 2

2

(

)

Check. 1 2 x + 25x + 9 (2x + 400) - 9918x - 3600 2 3 = x + 200x 2 + 50x 2 + 10, 000x + 18x + 3600

–––––––––––––––––––––

(_

+ 25x + 9 ⟌ x 3 + 250x 2 + 100x + 0 -(x + 50x + 18x)

―――――――― 3

2

200x 2 + 82x + 0

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

―――――――――――

AVOID COMMON ERRORS Students 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.

401

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)

- 9918x - 3600 = x 3 + 250x 2 + 100x

-9918x - 3600

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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.

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

= 7x 3 - 56x 2 + 52x 2 - 416x + 16x - 128 + 28 = 7x 3 - 4x 2 - 400x - 100

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

0

-28.5

-9

10

-2

0.5

7

-0.5

8

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

――――――――――――― 8 -2 -28 -2 9.5 11.

12.

- 0.5 + 9.5 = 8x 4 - 28.5x 2 - 9x + 10

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

A 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.

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

7 -4 -400 -100 56 416 128 ――――――――― 7 52 16 28

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

-0.25

AVOID COMMON ERRORS

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

6 -5.5 -10 -2.5 -3.5 9 ――――――――― 2.5 3.5 -9 -1 2.5

= 2.5x 3 + 2.5x 2 + 3.5x 2 + 3.5x - 9x - 9 - 1 = 2.5x 3 + 6x 2 - 5.5x - 10

(-25x 4 - 247x 3 + 50x 2 + 200x + 10) ÷ (x + 10) -10

Determine whether the given binomial is a factor of the polynomial p(x). If so, find the remaining factors of p(x). 13. p(x) = x 3 + 2x 2 - x - 2; (x + 2)

-2

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

-2

1 2 -1 -2 -2 0 2 ―――――― 1 0 -1 0

x + 2 is a factor.

6 0 -5 -10 -4 -4 8 -6 ―――――――――― 2 2 -4 3 -16 2

x + 2 is not a factor.

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

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3 2 -25 -247 50 200 10 (x + 10)(-25x + 3x + 20x) + 10 4 3 3 2 2 250 -30 -200 0 ―――――――――――― = -25x -250x + 3x + 30x + 20x + 200x + 10 -25 3 20 0 10 = -25x 4 - 247x 2 + 50x 2 + 200x +10

2

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

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

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Dividing Polynomials 402

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

INTEGRATE MATHEMATICAL PROCESSES Focus on Math Connections

8

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.

1 -22 157 -360 8 -112 360 ―――――――――― 1 -14 45 0 - 8 is a factor.

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

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

16. p(x) = 3x 4 - 6x 3 + 6x 2 + 3x - 30; (x - 2)

2

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

3 -6 6 3 -30 6 0 12 30 ――――――――― 3 0 6 15 0 x - 2 is a factor.

3x 3 + 6x + 15 = 3(x 3 + 2x + 5)

So, p(x) = 3x 4 - 6x 3 + 6x 2 + 3x - 30 = 3(x 3 + 2x + 5)(x - 2).

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

3

4 -12 2 -5 12 0 6 ―――――――― 4 0 2 1 x - 3 is not a factor.

18. The volume of a rectangular prism is modeled by the function V(x) = x 3 - 8x 2 + 19x - 12. Given V(1) = 0 and V(3) = 0 , identify the other value of x for which V(x) = 0, which will give the missing dimension of the prism.

© Houghton Mifflin Harcourt Publishing Company

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. V(4) = 0 19. 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 top face of the prism.

-2

1

-1

-6

0

-2

6

0

―――――――― 1 -3 0 0

So, the area can be represented by x 2 - 3x. Module 7

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20. 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) = 0.5t 3 + 6t 2 + 10t. Write an expression that represents the voltage of the system. Recall that V = __Pl .

-2

INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning 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.

0.5 6 10 0 -1 -10 0 ――――――― 0.5 5 0 0

The voltage can be represented by 0.5t 2 + 5t. 21. Geometry The volume of a hexagonal pyramid is modeled by the function 2 V(x) = __13 x 3 + __43 x + __23 x - __13 . Given the height x + 1, use polynomial division to find an expression for the area of the base. (Hint: For a pyramid, V = __13 Bh.)

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

-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.

A. 2

3 0 -2 -8 6 12 20 ―――――― 3 6 10 12

© Houghton Mifflin Harcourt Publishing Company • ©Ted Kinsman/Science Photo Library

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

3 0 -2 -8 -6 12 -20 ――――――― 3 -6 10 -28

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

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PEERTOPEER DISCUSSION

H.O.T. Focus on Higher Order Thinking

23. Multi-Step Use synthetic division to divide p(x) = 3x 3 - 11x 2 - 56x - 50 by (3x + 4). Then check the solution. 4 Rewrite 3x + 4 as x + . 3 3 -11 -56 -50 -4 3 -4 20 48 ――――――――― 3 -15 -36 -2

Instruct 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.

(

_

_)

The quotient needs to be divided by 3. 3x - 15x - 36 __ =x 2

3

Check.

JOURNAL

(

2

- 5x - 12

)

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

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

3

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

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

= 3x 3 - 11x 2 - 56x - 50 24. Critical Thinking The polynomial ax 3 + bx 2 + cx + d is factored as 3(x - 2)(x + 3)(x - 4). What are the values of a and d? Explain.

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a = 3; d = 72; The value of a is the leading coefficient, 3; the value of d is the product of the constant terms of each factor and the leading coefficient 3. 25. 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

Yes

Yes

Yes

Yes

Subtraction

No

Yes

Yes

Yes

Multiplication

Yes

Yes

Yes

Yes

Division (by nonzero)

No

No

Yes (nonzero)

Yes (nonzero)

Polynomials are similar to rational numbers with respect to closure. They are closed under each operation if division is nonzero. Module 7

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Lesson Performance Task

AVOID COMMON ERRORS

The table gives the attendance data for all divisions of NCAA Women’s Basketball.

Students 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.

NCAA Women’s Basketball Attendance Attendance Number of teams (in thousands) for in all 3 divisions all 3 divisions 1003 10,878.3

2006–2007

Years since 2006–2007 0

2007–2008

1

1013

11,120.8

2008–2009

2

1032

11,160.3

Season

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 polynomial regression on the data pairs (t, T) and (t, A) where t = years since the 2006–2007 season, T = number of teams, and A = attendance (in thousands). For each set of data pairs, choose the regression model having the least degree that best fits the data.

INTEGRATE MATHEMATICAL PROCESSES Focus on Reasoning

A(t)

. Carry out the division Then create a model for the average attendance per team: A avg(t) = ___ T(t) remainder to write A avg(t) in the form quadratic quotient +  _______ . () Tt

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.

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) = 13.80t 3 - 121.6t 2 + 329.8t + 10,880 Online computer algebra system result:

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© Houghton Mifflin Harcourt Publishing Company

12,981,600 A avg(t) = 1.30558t 2 - 135.64t + 12,927.9 - __ 10.57t + 1005

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Have students research the attendance for a specific NCAA women’s basketball team for a single season. Have them compare that number 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.

2/20/14 9:31 PM

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.

Dividing Polynomials 406