7.1 Finding Rational Solutions of Polynomial Equations

(1)

Name Class Date

Explore Relating Zeros and Coefficients of

Polynomial Functions

The zeros of a polynomial function and the coefficients of the function are related. Consider the polynomial function ƒ (x) = (x + 2) (x -1) (x + 3) .

A

Identify the zeros of the polynomial function.

B

Multiply the factors to write the function in standard form.

C

How are the zeros of ƒ (x) related to the standard form of the function?

D

Now consider the polynomial function g (x) = (2x + 3)(4x - 5) (6x - 1) . Identify the zeros of this function.

E

Multiply the factors to write the function in standard form.

F

How are the zeros of g (x) related to the standard form of the function?

Resource Locker

The zeros are x = -2, x = 1, and x = -3.

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

)

=

(

^x²+ 2x - x - 2

)

(x + 3

)

=

(

^x²+ x - 2

)

(^x+ 3

)

= x ³ + 3 x ² + x ² + 3x - 2x - 6

= x ³ + 4 x ² + x - 6

Each of the zeros of the polynomial function is a factor of the constant term in the standard form.

The zeros are x = - ³

_

₂ , x = ⁵

_

₄ , and x = ¹

_

₆ .

g (x) = (2x + 3)⁽4x - 5) (6x - 1)

=

(

^{8 x}²- 10x + 12x - 15

)

(^6x- 1)

=

(

^{8 x}²^{+ 2x - 15}

)

(6x - 1)

= 48 x ³ - 8 x ² + 12 x ² - 2x - 90x + 15

= 48 x ³ + 4 x ² - 92x + 15

Each of the numerators of the zeros is a factor of the constant term, 15, and each of the denominators is a factor of the leading coefficient, 48.

Module 7 341 Lesson 1

7.1 Finding Rational Solutions

of Polynomial Equations

Essential Question: How do you find the rational roots of a polynomial equation?

A2_MNLESE385894_U3M07L1.indd 341 3/19/14 2:37 PM

Common Core Math Standards

The student is expected to:

A-APR.2

Know and apply the Remainder Theorem: For a polynomial p(x) and a

number a, the remainder on division by x – a is p(a), so p(a) = 0 if and

only if (x – a) is a factor of p(x). Also A-APR.3, A-CED.3

Mathematical Practices

MP.2 Reasoning

Language Objective

Explain to a partner how to identify the factors of a polynomial function.

HARDCOVER PAGES 247254

Turn to these pages to

find this lesson in the

hardcover student

edition.

Finding Rational

Solutions of

Polynomial Equations

ENGAGE

Essential Question: How do you find

the rational roots of a polynomial

equation?

Use the Rational Root Theorem to identify possible

rational roots. Check each by using synthetic

substitution. If a rational root is found, repeat the

process on the quotient obtained from the bottom

row of the synthetic substitution. Continue to find

rational roots in this way until the quotient is

quadratic, at which point you can try factoring to

identify the last two rational roots.

PREVIEW: LESSON

PERFORMANCE TASK

View the Engage section online. Discuss the photo

and how the number of tourists in any given year can

vary depending on many factors. Then preview the

Lesson Performance Task.

341

HARDCOVER

Turn to these pages to

find this lesson in the

hardcover student

edition.

Comp

any

Name

Class Date

Explore Relating Zeros and Coefficients of Polynomial Functions The zeros of a polynomial function and the coefficients of the function are related. Consider the polynomial function ƒ ⁽x) = (x + 2) (x -1) (x + 3) .

 Identify the zeros of the polynomial function.

 Multiply the factors to write the function in standard form.

 How are the zeros of ƒ (x) related to the standard form of the function?

 Now consider the polynomial function g (x) = (2x + 3) (4x - 5) (6x - 1) . Identify the zeros of this function.

 Multiply the factors to write the function in standard form.

 How are the zeros of g (x) related to the standard form of the function?

Resource Locker

A-APR.2 Know and apply the Remainder Theorem: For a polynomial p (x) and a number a, the remainder on division by x - a is p (a) , so p (a) = 0 if and only if (

x - a) is a factor of p (x) . Also

A-APR.3, A-CED.3

The zeros are x = -2, x = 1, and x = -3.

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

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

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

= x ³ + 3 x ² + x ² + 3x - 2x - 6

= x ³ + 4 x ² + x - 6

Each of the zeros of the

polynomial function is a factor of the constant term in the standard form.

The zeros are x = - 3 _ 2 , x = 5 _ 4 , and x = 1 _ 6 . g (x) = (2x + 3) (4x - 5) (6x - 1)

= (8 x ² - 10x + 12x - 15) (6x - 1)

= (^{8 x}² + 2x - 15) (6x - 1)

= 48 x ³ - 8 x ² + 12 x ² - 2x - 90x + 15

= 48 x ³ + 4 x ² - 92x + 15

Each of the numerators of the zeros is a factor of the constant term, 15, and each of the denominators is a factor of the leading coefficient, 48.

Module 7 341

Lesson 1

7 . 1 Finding Rational Solutions of Polynomial Equations

Essential Q

uestion: How do you find the rational roots of a polynomial equation?

DO NOT EDIT--Changes must be made through “File info”

CorrectionKey=NL-A;CA-A

A2_MNLESE385894_U3M07L1.indd 341

3/19/14 2:39 PM

7 . 1

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1. In general, how are the zeros of a polynomial function related to the function written in standard form?

2. Discussion Does the relationship from the first Reflect question hold if the zeros are all integers? Explain.

3. If you use the zeros, you can write the factored form of g (x) as g (x) =

(

^{x +}^__³²

)

(

^{x -}^__⁵⁴

)

(

^{x -}^__¹⁶

)

, rather than as g (x) = (^{2x + 3}) (4x - 5) (6x - 1) . What is the relationship of the factors between the two forms?

Give this relationship in a general form.

Explain 1 Finding Zeros Using the Rational Zero Theorem

If a polynomial function p (x) is equal to (â¹ x + b 1)(â² x + b 2)(â³ x + b 3) , where a 1 , a 2 , a 3 , b 1 , b 2 , and b 3

are integers, the leading coefficient of p (x) will be the product a 1 a 2 a 3 and the constant term will be the product b 1 b 2 b 3 . The zeros of p (x) will be the rational numbers - __ a ^b¹1 , - __ ^b a ²2 , - __ ^b a ³3 .

Comparing the zeros of p (x) to its coefficient and constant term shows that the numerators of the polynomial’s zeros are factors of the constant term and the denominators of the zeros are factors of the leading coefficient. This result can be generalized as the Rational Zero Theorem.

Rational Zero Theorem

If p (x) is a polynomial function with integer coefficients, and if m _ n is a zero of p ⁽x)

(

^p

⁽

^m^__ⁿ

⁾

^{= 0}

)

^,

then m is a factor of the constant term of p (x) and n is a factor of the leading coefficient of p(x).

Example 1 Find the rational zeros of the polynomial function; then write the function as a product of factors. Make sure to test the possible zeros to find the actual zeros of the function.



ƒ (x) = x ³ + 2 x ² - 19x - 20

a. Use the Rational Zero Theorem to find all possible rational zeros.

Factors of -20: ±1, ±2, ±4, ±5, ±10, ±20 b. Test the possible zeros. Use a synthetic division

table to organize the work. In this table, the first row (shaded) represents the coefficients of the polynomial, the first column represents the divisors, and the last column represents the remainders.

m

_

_n 1 2 -19 -20

1 1 3 -16 -36

2 1 4 -11 -42

4 1 6 5 0

5 1 7 16 60

Each of the numerators of the zeros is a factor of the constant term. Each of the denominators of the zeros is a factor of the leading coefficient.

Yes; If the zeros are all integers, each of them can be written with a denominator of 1. Each of the numerators is still a factor of the constant term.

In each factor, the denominator of the fraction becomes the coefficient of the variable.

In general, if the zero is - b

_

_a , the factor can be written as

(

^{ax + b}

)

^.

A2_MNLESE385894_U3M07L1 342 16/10/14 10:14 AM

Integrate Mathematical Practices

This lesson provides an opportunity to address Mathematical Practice MP.2,

which calls for students to translate between multiple representations and to

“reason abstractly and quantitatively.” Students explore the relationship between

the factors of a polynomial function and its zeros. They learn how to identify the

zeros given the factors, and the factors given the zeros. They then explore the

relationships between the rational zeros of a function and its leading coefficient

EXPLORE

Relating Zeros and Coefficients of

Polynomial Functions

INTEGRATE TECHNOLOGY

Students have the option of completing the Explore

activity either in the book or online.

QUESTIONING STRATEGIES

What is the relationship between the factors of

a polynomial function and the zeros of the

function? The zeros are the values of x found by

setting each factor equal to 0 and solving for x.

If a zero of a polynomial function is 7 ___ 13 , what

do you know about the coefficients when the

polynomial is written in standard form? 7 is a factor

of the constant term and 13 is a factor of the leading

coefficient.

EXPLAIN 1

Finding Zeros Using the Rational Zero

Theorem

QUESTIONING STRATEGIES

Is every zero of a polynomial function

represented in the set of numbers given by the

Rational Zero Theorem? No. The Rational Zero

Theorem gives only those zeros that are rational

numbers. A polynomial function can also have zeros

that are irrational numbers or imaginary numbers.

PROFESSIONAL DEVELOPMENT

(3)

c. Factor the polynomial. The synthetic division by 4 results in a remainder of 0, so 4 is a zero and the polynomial in factored form is given as follows:

(x - 4)

(

^x²^{+ 6x + 5}

)

^{= 0}

(x - 4) (x + 5) (^{x + 1})^{= 0} x = 4, x = -5, or x = -1 The zeros are x = 4, x = -5, and x = -1.

B

ƒ (x) = x ⁴ - 4 x ³ - 7 x ² + 22x + 24

a. Use the Rational Zero Theorem to find all possible rational zeros.

Factors of 24: ± , ± , ± , ± , ± , ± , ± , ± b. Test the possible zeros. Use a synthetic division table.

m

_

_n ¹ -4 -7 22 24

1 2 3

c. Factor the polynomial. The synthetic division by results in a remainder of 0, so is a zero and the polynomial in factored form is given as follows:

(x - ) ( x ³ - x ² - x - ) = 0

d. Use the Rational Zero Theorem again to find all possible rational zeros of g (x) = x ³ - x ² - x - .

Factors of -8: ± , ± , ± , ± e. Test the possible zeros. Use a synthetic division table.

m

_

_n ¹ -1 -10 -8

1 2 4

f. Factor the polynomial. The synthetic division by results in a remainder of 0, so is a zero and the polynomial in factored form is:

(x - ) (x - )

(

^x² + x +

)

^{= 0}

(x - ) (x - ) (x + ) (x + ) = 0 x = , x = , x = , or x = The zeros are .

1 2 3 4 6 8 12 24

1 -3 -10

-11 0

12

-8 0

24 36

-10 -2 -1 1

1

3 3

3 10 8

8 10

1 2 4 8

1 3 2 0

1 1 -8 -24

1 0 -10 -18

4 4

3 4 1 3 2

1 -2

x = 3, x = 4, x = -2, and x = -1 2

4 3

3 4 -1

A2_MNLESE385894_U3M07L1.indd 343 7/7/14 10:24 AM

COLLABORATIVE LEARNING

Small Group Activity

Have students work in groups of 3–4 students. Instruct each group to create a

fifth-degree polynomial function with rational zeros, not all of which are integers.

Ask them to write their functions in standard from. Have groups exchange

functions, and have each group create a poster showing how to apply the Rational

Zero Theorem to find the zeros of the function. Students’ posters should also

show verification that each number is indeed a zero of the function.

AVOID COMMON ERRORS

Some students may forget to include 1 and –1 in their

list of possible rational zeros. You may want to

suggest that they write these first so that they are not

inadvertently left off the list.

QUESTIONING STRATEGIES

If the leading coefficient of a polynomial

function with integer coefficients is 1, what

can you conclude about the function’s rational zeros?

Explain your reasoning. They must be integers,

because when you apply the Rational Zero

Theorem, n can equal only 1 or –1 in m ___ _{n .}

(4)

4. How is using synthetic division on a 4 ^th degree polynomial to find its zeros different than using synthetic division on a 3 ^rd degree polynomial to find its zeros?

5. Suppose you are trying to find the zeros the function ƒ (x) = x ² + 1. Would it be possible to use synthetic division on this polynomial? Why or why not?

6. Using synthetic division, you find that __ 1 ₂ is a zero of ƒ (x) = 2 x ³ + x ² - 13x + 6. The quotient from the synthetic division array for ƒ

(

¹^__²

)

^{is 2 x}² + 2x - 12. Show how to write the factored form of ƒ (x) = 2 x ³ + x ² - 13x + 6 using integer coefficients.

Your Turn

7. Find the zeros of ƒ (x) = x ³ + 3 x ² - 13x- 15.

To find the zeros of a 4 ^th degree polynomial using synthetic division, you need to use synthetic division to reduce that polynomial to a 3 ^rd degree polynomial and then use synthetic division again to reduce that polynomial to a quadratic polynomial that can be factored, if possible.

It would not be possible to find the zeros of this polynomial using synthetic substitution because the function has no rational roots, only complex roots.

Using

__

1 ₂ as a zero and the quotient 2 x ² + 2x - 12 you can write f (x) = 2 x ³ + x ² - 13x + 6 as f (x) =

(

^{x - 1}^_²

)

⁽

^{2 x}² + 2x - 12

)

^.

f (x) =

(

^{x - 1}^_²

)

⁽

^{2 x}² + 2x - 12

)

⁼

(

^{x - 1}

^_

²

)

⁽²⁾

⁽

^x²^{+ x - 6}

⁾

= (2x - 1)

(

^x²^{+ x - 6}

)

⁼⁽^{2x - 1}⁾(x + 3)(x - 2)

a. Use the Rational Zero Theorem. Factors of -15: ±1, ±3, ±5, ±15 b. Test the possible zeros to find one that is actually a zero.

m

_

_n ¹ -3 -13 -15

1 1 4 -9 -24

3 1 6 5 0

c. Factor the polynomial using 3 as a zero.

(x - 3)

(

^x² + 6x + 5

)

= 0 (x - 3) (^x+ 1)(^x+ 5)= 0

x = 3, x = -1, or x = -5 The zeros are x = 3, x = -1, and x = -5.

DIFFERENTIATE INSTRUCTION

Visual Cues

Encourage students to circle the leading coefficient in the function and to write

“n is a factor of” above it, and to circle the constant term in the function and to

write “m is a factor of” above it. This will be helpful when applying the Rational

Zero Theorem, and will keep students from erroneously writing the reciprocals of

the possible rational zeros, especially since the usages of m and n appear in reverse

alphabetical order with respect to the function.

INTEGRATE MATHEMATICAL

PRACTICES

Focus on Math Connections

MP.1 Remind students that a zero of a function is a

number from the domain that the function pairs with

0. Discuss that, for this reason, a graph of the

function will have an x-intercept at each zero.

Students can then make a concrete connection

between the rational zeros they identify for a

function, and the role the zeros play in the graph of

the function.

(5)

Explain 2 Solving a Real-World Problem Using

the Rational Root Theorem

Since a zero of a function ƒ (x) is a value of x for which ƒ (x) = 0, finding the zeros of a polynomial function p (x) is the same thing as find the solutions of the polynomial equation p (x) = 0. Because a solution of a polynomial equation is known as a root, the Rational Zero Theorem can be also expressed as the Rational Root Theorem.

Rational Root Theorem

If the polynomial p (x) has integer coefficients, then every rational root of the polynomial equation p (x) = 0 can be written in the form m __ n , where m is a factor of the constant term of p (x) and n is a factor of the leading coefficient of p (x) .

Engineering A pen company is designing a gift container for their new premium pen. The marketing department has designed a pyramidal box with a rectangular base. The base width is 1 inch shorter than its base length and the height is 3 inches taller than 3 times the base length. The volume of the box must be 6 cubic inches. What are the dimensions of the box? Graph the volume function and the line y = 6 on a graphing calculator to check your solution.

A. Analyze Information

The important information is that the base width must be inch shorter than

the base length, the height must be inches taller than 3 times the base length,

and the box must have a volume of cubic inches.

B. Formulate a Plan

Write an equation to model the volume of the box.

Let x represent the base length in inches. The base width is and the height is , or .

1 _ 3 ℓw h = V 1

_

3 ( ) ⁽x - ) (3) (x + ) = x ³ - x - = 0

History in the mark ing

1

x - 1 3x + 3 3 (x + 1)

3

6

x 1 1

1

1 6

6

A2_MNLESE385894_U3M07L1 345 6/28/14 2:13 PM

EXPLAIN 2

Solving a Real-World Problem Using

the Rational Root Theorem

CONNECT VOCABULARY

Explain how the words zeros and roots (or solutions)

have similar meanings but are used in different

contexts. The zeros of a function are the roots (or

solutions) of the related equation.

QUESTIONING STRATEGIES

Why is it necessary to rewrite the equation so

that it is equal to 0? In order to find the roots

of an equation using the Rational Root Theorem, the

equation must be in the form p

(

x

)

= 0.

What information is obtained by applying the

Rational Zero Theorem to a polynomial

function? A list of all possible rational zeros of the

function

(6)

C. Solve

Use the Rational Root Theorem to find all possible rational roots.

Factors of -6: ± , ± , ± , ± Test the possible roots. Use a synthetic division table.

m

_

_n ¹ ⁰ -1 -6

1

2

3

Factor the polynomial. The synthetic division by results in a remainder of 0, so is a root and the polynomial in factored form is as follows:

(

x -

)

(

^x²+ x +

)

^{= 0}

The quadratic polynomial produces only roots, so the only possible answer for the base length is inches. The base width is inch and the height is inches.

D. Justify and Evaluate

The x-coordinates of the points where the graphs of two functions, f and g, intersect is the solution of the equation f (x) = g (x) . Using a graphing calculator to graph the volume function and y = 6 results in the graphs intersecting at the point . Since the x-coordinate is , the answer is correct.

Your Turn

8. Engineering A box company is designing a new rectangular gift container. The marketing department has designed a box with a width 2 inches shorter than its length and a height 3 inches taller than its length.

The volume of the box must be 56 cubic inches. What are the dimensions of the box?

1 2 3

1 1 0 -6

1 2 3 0

1 3 8 18

6

2

complex

1 2

9 2

2 1 2 3

1

(2, 6) 2

A. The box width must be 2 inches shorter than the length, the height must be 3 inches taller than the width, and the box must have a volume of 56 cubic inches.

B. Let x represent the length in inches. The width is x - 2 and the height is x + 3.

ℓwh = V (x) (x - 2) (x + 3) = 56

x ³ + x ² - 6x = 56 x ³ + x ² - 6x - 56 = 0

A2_MNLESE385894_U3M07L1 346 6/27/14 10:32 PM

INTEGRATE MATHEMATICAL

PRACTICES

Focus on Critical Thinking

MP.3 Prompt students to recognize that any

rational roots found by factoring the resulting

quadratic polynomial must be numbers that were

identified as possible rational roots initially. This may

help them to catch errors in factoring, or in

performing the synthetic division.

(7)

Elaborate

9. For a polynomial function with integer coefficients, how are the function’s coefficients and rational zeros related?

10. Describe the process for finding the rational zeros of a polynomial function with integer coefficients.

11. How is the Rational Root Theorem useful when solving a real-world problem about the volume of an object when the volume function is a polynomial and a specific value of the function is given?

12. Essential Question Check-In What does the Rational Root Theorem find?

C. Use the Rational Root Theorem. Factors of -56: ±1, ±2, ±4, ±7, ±8, ±14, ±28, ±56 Test the possible roots to find one that is actually a root. Use a synthetic division table.

p

_

_q ¹ ¹ -6 -56

1 1 2 -4 -60

2 1 3 0 -56

4 1 5 14 0

Factor the polynomial. using 4 as a root.

(x - 4)

(

^x²+ 5x + 14

)

= 0

The quadratic polynomial produces only complex roots. The only possible answer for the length is 4 inches. The width is 2 inches and the height is 7 inches.

D. Using a graphing calculator, the graphs intersect at

(

^{4, 56}

)

, which validates the answer.

The rational zeros of a polynomial function with integer coefficients are in the form

__

m _n , where m is a factor of the constant term and n is a factor of the leading coefficient.

Using the Rational Zero Theorem to find all possible rational zeros, test the possible zeros to find one that is actually a zero by using a synthetic division table to organize the work and factor the polynomial.

The theorem is useful in this case because it allows you to find the rational roots of the polynomial equation created when you set the volume function equal to the given value.

By rewriting the equation so that one side is 0, you can use the Rational Root Theorem to find the dimension given by the variable and then find the other dimensions.

The Rational Root Theorem finds the possible rational roots of a polynomial equation.

A2_MNLESE385894_U3M07L1 347 6/27/14 10:32 PM

LANGUAGE SUPPORT

Communicating Math

Have students work in pairs. Instruct one student to write a polynomial function

in factor form. Have the second student identify the zeros of the function and

explain why they are the zeros. The students switch roles and repeat the process.

Repeat the example from the lesson to provide a format.

ELABORATE

INTEGRATE MATHEMATICAL

PRACTICES

Focus on Technology

MP.5 Have students discuss how they could use a

graphing utility to help determine which numbers

from their list of possible rational zeros are more

likely than others to be zeros. Students should

recognize that they can use the x-intercepts of the

graph to help them focus in on which numbers on

their lists are good candidates to test as possible

zeros.

QUESTIONING STRATEGIES

If a cubic function has only one rational root,

what will be true about the quadratic

polynomial quotient that results from synthetic

division by the rational root? It will not be

factorable over the set of integers.

SUMMARIZE THE LESSON

How can you use the Rational Root Theorem

to find the rational solutions of a polynomial

equation? You can write the equation in the form

p

₍

x

₎

= 0, and then use the theorem to identify

possible roots of the equation. These roots will be of

the form

_{__}

p

q . You can then test the possible roots

using synthetic substitution. If you can reduce the

polynomial to a quadratic, you can try factoring the

quadratic to find any other rational roots.

(8)

Find the rational zeros of each polynomial function. Then write each function in factored form.

1. ƒ (x) = x ³ − x ² − 10x − 8 2. ƒ (x) = x ³ + 2 x ² - 23x - 60

3. j (x) = 2 x ³ - x ² - 13x - 6 4. g (x) = x ³ - 9 x ² + 23x − 15

5. h (x) = x ³ - 5 x ² + 2x + 8 6. h (x) = 6 x ³ - 7 x ² - 9x − 2

7. s (x) = x ³ - x ² − x + 1 8. t (x) = x ³ + x ² − 8x − 12

• Online Homework

• Hints and Help

• Extra Practice

Evaluate: Homework and Practice

Factors of −8 : ±1, ±2, ±4, ±8 Factors of −60 : ±1, ±2, ±3, ±4, ±5, ±6,

±10, ±12, ±15, ±20, ±30, ±60 (x - 4)

(

^x² + 3x + 2

)

= 0

(x - 4) (x + 2)(^{x + 1})^{= 0} x = 4, x = -2, or x = -1 f (x) = (x - 4) (x + 2)(x + 1)

(x - 5)

(

^x² + 7x + 12

)

= 0 (x - 5) (x + 3)(x + 4) = 0 x = 5, x = -3, or x = -4 f (x) = (x - 5) (x + 3)(x + 4) Factors of −15 : ±1, ±3, ±5, ±15 Factors of −6 : ±1, ±2, ±3, ±6

4 is a zero.

5 is a zero.

(x - 1)

(

^x² - 8x + 15

)

^{= 0}

(x - 1) (x -5) (x - 3)= 0 x = 1, x = 5, or x = 3 g (x) = (x - 1) (x - 5) (x - 3) (x - 3)

(

^{2 x}²^{+ 5x + 2}

)

^{= 0}

(x - 3) (2x + 1)(x + 2) = 0 x = 3, x = - ¹

_

₂^{, or x}^{= -2}

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

3 is a zero. 1 is a zero.

Factors of 8 : ±1, ±2, ±4, ±8

(x - 2)

(

^x² - 3x - 4

)

= 0 (x - 2) (x - 4) (^x+ 1)= 0 x = 2, x = 4, or x = -1 h (x) = (x - 2) (x - 4) (x + 1)

Factors of −2 : ±1, ±2

(x - 2)

(

^6x²^{+ 5x + 1}

)

^{= 0}

(x - 2) (2x + 1)(3x + 1) = 0 x = 2, x = -

_

¹₂ , or x = -

_

¹₃

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

Factors of 1 : ±1 Factors of −12 : ±1, ±2, ±3, ±4, ±6, ±12

(x - 3)

(

^x²^{+ 4x + 4}

)

^{= 0}

(x - 3) (x + 2)(x + 2) = 0 x = 3 or x = -2

t (x) = (^x- 3)(^x+ 2)(^x+ 2) (x - 1)

(

x ² - 1

)

= 0

(x - 1) (x + 1)⁽x - 1) = 0 x = 1 or x = -1

s (x) = (x - 1) (^x+ 1)⁽^x- 1)

A2_MNLESE385894_U3M07L1 348

Exercise Depth of Knowledge (D.O.K.) Mathematical Practices

16/10/14 10:37 AM

1–12

1

Recall of Information

MP.5

Using Tools

13–17

2

Skills/Concepts

MP.4

Modeling

18–19

2

Skills/Concepts

MP.3

Logic

20

3

Strategic Thinking

MP.2

Reasoning

21

3

Strategic Thinking

MP.3

Logic

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

Explore

Relating Zeros and Coefficients of Polynomial Functions

Exercise 17

Example 1

Finding Zeros Using the Rational Zero Theorem

Exercises 2–12

Example 2

Solving a Real-World Problem Using the Rational Root Theorem

Exercises 13–16

INTEGRATE MATHEMATICAL

PRACTICES

Focus on Patterns

MP.8 Students can use patterns in the signs of the

terms in the polynomial function to help them decide

which of the possible rational zeros to test. For

example, if the signs of the terms in the polynomial

function (or in the quotient after dividing

synthetically) are all positive, students need not check

any positive numbers on their lists.

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9. k (x) = x ⁴ + 5 x ³ - x ² − 17x + 12 10. g (x) = x ⁴ - 6 x ³ + 11 x ² - 6x

11. h (x) = x ⁴ - 2 x ³ - 3 x ² + 4x + 4 12. ƒ (x) = x ⁴ - 5 x ² + 4

13. Manufacturing A laboratory supply company is designing a new rectangular box in which to ship glass pipes. The company has created a box with a width 2 inches shorter than its length and a height 9 inches taller than twice its length. The volume of each box must be 45 cubic inches. What are the dimensions?

Factors of 12 : ±1, ±2, ±3, ±4, ±6, ±12

Factor the polynomial.

(x − 1)

(

^x³+ 6 x ² + 5x − 12

)

Factors of 12 : ±1, ±2, ±3, ±4, ±6, ±12

g (x) = x

(

^x³- 6 x ² + 11x - 6

)

Factors of -6 : ±1, ±2, ±3, ±6

(x - 1) (x - 1)

(

^x² + 7x + 12

)

= 0 (x - 1) (x - 1) (x + 3)(x + 4) = 0 x = 1, x = -3, or x = -4

k (x) = (x - 1) (x - 1) (x + 3)(x + 4)

(x) (x - 1)

(

^x²- 5x + 6

)

= 0 (x) (x - 1) (x - 3) (x - 2) = 0 x = 1, x = 0, x = 3, or x = 2 g (x) = (x) (x - 1) (x - 3) (x - 2) 1 is a zero.

1 is a zero.

Factors of 4 : ±1, ±2, ±4

(x − 2)

(

^x³ − 3x − 2

)

Factors of -2 : ±1, ±2

(x - 2) (x - 2)

(

^x² + 2x + 1

)

= 0 (x - 2) (x - 2) (^{x + 1})(^{x + 1})^{= 0} x = -1 or x = 2

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

f (x) = (x - 1)

(

^x³+ x ² − 4x − 4

)

Factors of −4 : ±1, ±2, ±4 Factors of 4 : ±1, ±2, ±4

(x - 1) (x - 2)

(

^x² + 3x + 2

)

= 0 (x - 1) (x - 2) (x + 2)(^{x + 1})^{= 0} x = 1, x = 2, x = -2, or x = -1 f (x) = (x - 1) (x - 2) (x + 2)(x + 1)

Let x represent the length in inches. Then the width is x - 2 and the height is 2x + 9.

ℓwh = V (x) (x - 2) (2x + 9) = 0 2 x ³ + 5 x ² - 18x = 45 2 x ³ + 5 x ² - 18x - 45 = 0

Factors of -45: ±1, ±3, ±5, ±9, ±15, ±45 3 is a root.

(x - 3)

(

^{2 x}² + 11x + 15

)

= 0, so (x - 3) (2x + 5)(x + 3) = 0 x = 3, x = ___ ^-52 , or x = -3

Length cannot be negative. The length must be is 3 inches. The width is 1 inch and the height is 15 inches.

A2_MNLESE385894_U3M07L1 349 16/10/14 10:46 AM

AVOID COMMON ERRORS

Students may incorrectly conclude that a polynomial

function that has n rational zeros has only n real

zeros. Explain that the function may have irrational

zeros as well, and irrational zeros are real zeros.

CONNECT VOCABULARY

Have students, in their own words, explain how the

Rational Zero Theorem and the Rational Root

Theorem are related (for example, a solution of a

polynomial equation is often called a root).

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14. Engineering A natural history museum is building a pyramidal glass structure for its tree snake exhibit. Its research team has designed a pyramid with a square base and with a height that is 2 yards more than a side of its base. The volume of the pyramid must be 147 cubic yards. What are the dimensions?

15. Engineering A paper company is designing a new, pyramid- shaped paperweight. Its development team has decided that to make the length of the paperweight 4 inches less than the height and the width of the paperweight 3 inches less than the height. The paperweight must have a volume of 12 cubic inches. What are the dimensions of the paperweight?

Let x represent the side of the square base in yards. The height is x + 2.

1

_

₃ ℓwh = V 1

_

3 (x) (x) (x + 2) = 147 1

_

₃

(

^x³+ 2 x ²

)

= 147 x ³ + 2 x ² = 441

x ³ + 2 x ² - 441 = 0 Factors of -441: ±1, ±3, ±7, ±9, ±21, ±49, ±63, ±147, ±441 7 is a root.

(x - 7)

(

^x² + 9x + 63

)

= 0

The quadratic factor produces only complex roots. So, each side of the base is 7 yards and the height is 9 yards.

Let x represent the height in inches. The length is x - 4 and the width is x - 3.

1

_

3 ℓwh = V 1

_

3 (x - 4) (x - 3)⁽^x⁾ = 12 1

_

3

(

^x³ - 7x ² + 12x

)

= 12 x ³ - 7 x ² + 12x = 36 x ³ - 7 x ² + 12x - 36 = 0

Factors of -36: ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36 6 is a root.

(x - 6)

(

^x²^{- x + 6}

)

^{= 0}

The quadratic factor produces only complex roots. So, the height is 6 inches, the length is 2 inches, and the width is 3 inches.

CRITICAL THINKING

Discuss with students why the Rational Root

Theorem works, by applying it to a quadratic

equation, such as 2 x

²

+ x - 15 = 0, and showing

how the process of solving the equation by factoring

focuses on the factors of p and q in a way that is

similar to the process of the Rational Root Theorem.

Focus students’ attention on how p is the product of

the first coefficients of the factors, and q is the

product of the constant terms of the factors.

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16. Match each set of roots with its polynomial function.

A. x = 2, x = 3, x = 4 ƒ (x) = (x + 2) (x + 4)

(

^{x – 3}^_²

)

B. x = –2, x = –4, x = 3 _ 2 ƒ (x) =

(

^{x – 1}^_²

)

(

^{x – 5}^_⁴

)

⁽

^{x + 7}^_³

⁾

C. x = 1 _ 2 , x = 5 _ 4 , x = – 7 _ 3 ƒ (x) = (x – 2) (x – 3) (x – 4) D. x = – 4 _ 5 , x = 6 _ 7 , x = 4 ƒ (x) =

(

^{x + 4}^_⁵

)

(

^{x – 6}^_⁷

)

^{(x – 4)}

17. Identify the zeros of ƒ (x) = (x + 3) (x - 4) (x - 3) , write the function in standard form, and state how the zeros are related to the standard form.

H.O.T. Focus on Higher Order Thinking

18. Critical Thinking Consider the polynomial function g (x) = 2 x ³ - 6 x ² + πx + 5. Is it possible to use the Rational Zero Theorem and synthetic division to factor this polynomial?

Explain.

19. Explain the Error Sabrina was told to find the zeros of the polynomial function h (x) = x (x - 4) (^{x + 2}) . She stated that the zeros of this polynomial are x = 0, x = -4, and x = 2. Explain her error.

20. Justify Reasoning If c _ b is a rational zero of a polynomial function p (x) , explain why bx - c must be a factor of the polynomial.

21. Justify Reasoning A polynomial function p (x) has degree 3, and its zeros are –3, 4, and 6. What do you think is the equation of p (x) ? Do you think there could be more than one possibility? Explain.

B C A D

The zeros of f (x) are x = -3, x = 4, and x = 3.

f (x) = (x + 3)(x - 4)( x - 3) =

(

^x² + 3x - 4x - 12

)

(x - 3)

=

(

^x²- x - 12

)

(^x- 3)= x ³ - 3 x ² - x ² + 3x - 12x + 36

= x ³ - 4 x ² - 9x + 36

The zeros of f (x) are all factors of the constant term in the polynomial function.

No; it is not possible because the function contains a term, πx, whose coefficient is irrational and, therefore, not an integer.

For any factor

(

ax + b

)

, a zero occurs at -

_

^b_a . Sabrina forgot to include the negative sign when converting from her factors to the zeros.

Since p

(

^_

^c^b

)

= 0, x -

_

^c_b is a factor of p (x) by the Factor Theorem. So, p (x) =

(

^{x -}

^_

^b^c

)

^q⁽^x⁾^{and p}⁽^x⁾⁼^b

^_

^b

(

^{x -}

^_

^c^b

)

^q⁽^x⁾⁼^b¹

^_

⁽^{bx - c}⁾^q⁽^x⁾^{, which}

shows that bx - c is a factor of p (x) .

p (x) = (x + 3)⁽x - 4) (x - 6) ; any constant multiple of p (x) will also have degree 3 and the same zeros, so the equation can be any function of the form p (x) = a (x + 3)⁽x - 4) (x - 6) where a ≠ 0.