Textbook 7 answers worksheet PDF

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I want you guys to pass the algebra 2 class and I hope this helps...

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LESSON

7.1 Name

Finding Rational Solutions of Polynomial Equations

7.1

Finding Rational Solutions of Polynomial Equations

Explore

The student is expected to: A-APR.B.2

Relating Zeros and Coefficients of Polynomial Functions

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

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.B.3, A-CED.A.3

Mathematical Practices

Identify the zeros of the polynomial function.

B

Multiply the factors to write the function in standard form. f ( x) = ( x + 2) (x - 1)(x + 3 )

MP.2 Reasoning

= ( x2 + 2x - x - 2) (x + 3 )

Language Objective

= ( x2 + x - 2) (x + 3)

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

= x 3 + 3x2 + x 2 + 3x - 2x - 6



ENGAGE

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.



Multiply the factors to write the function in standard form. g( x) = ( 2x + 3 ) (4x - 5 ) (6x - 1 )

= (8x 2 - 10x + 12x - 15)( 6x - 1) = (8x 2 + 2x - 15)( 6x - 1) = 48x 3 - 8x2 + 12x 2 - 2x - 90x + 15



= 48x 3 + 4x2 - 92x + 15 How are the zeros of g( x) related to the standard form of the function? Each of the numerators of the zeros is a factor of the constant term, 15, a denominators is a factor of the leading coefficient, 48.

Module 7 Da te Cla s s

lut ions tio nal So Finding Ra ial Equatio ns of Po lynom

Na me

7.1

l equa tion? a polynomia l roots of r a, the the ra tiona mial ( p x) and a numbe . Als o do you find m: For a polyno p x( ) s t ion: How der Theore a factor of ia l Q ue the Remain ( x- a) is Es s e nt and apply and only if B.2 Know p , s o p( a) = 0 if COMMON A-APR. by x- a is ( a) CORE on divis ion remainder A-CED.A.3 A-APR.B.3,

nt s o f Co ef f icie Z ero s and ns the Rela t ing l Fun ct io . Consider n are related Po lyno mia of the functio

Exp lo re

coefficients n and the .) mial functio () x + 3 2)( of a polyno x ) ( x + x -1 The zeros function ƒ ( = polynomial

I dentify the



zeros of the

factors Multiply the (

B

= 1, a nd

x = -3.

rd form.

+x2 - x + ) x - 2() x 3 of the 2 = x( + x -2x -6 the z e ros 2 + x +3 Ea c h of = x 3 +3 x n? the functio f orm. 2 rd form of +x -6 ta nda rd the standa in the s =3 x+4 x related to nt te rm c ons ta of ƒ x( ) the zeros tor of the H ow are is a f a c the zeros l f unc tion () 6x - 1.)I dentify ( 4x - 5 polynomia 2( x + 3) ) x( = ng mial functio polyno the r Now conside _1. 5 n. _3 -x = _4, a nd x = 6 of this functio a re x2= , The z e ros rd form. n in standa write the functio factors to ) 1 Multiply the ) (4x - 5 ) (6 x x =( 2x + 3 x -1 ) - 15 )( 6 g () 2 - x 10 +12x = (8x )1 x ( 2 - 15 ) 6 2x 15 90 x + 2 - x = (8x + 2 3 2x +12x the 8 e a c h of = 48x n? 15, a nd + 15 the functio 2 -92x nt te rm, rd form of 3 4x c ons ta the standa = 48x + related to tor of the is a f a c of gx( ) the zeros the z e ros nt, 48. H ow are ra tors of oe f f ic ie a ding c the nume Lesson 1 Ea c h of of the le = x(

2



-2, x a re x = The z e ros

Re s ource Locke r

function. polynomial

standa function in to write the ) )( x + 3

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

 C o mp any

quadratic, at which point you can try factoring to identify the last two rational roots.

5 _1. The zeros are x = -_3, x = _ 4, and x = 6 2

Pub lishing

row of the synthetic substitution. Continue to find rational roots in this way until the quotient is

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



Harco urt

substitution. If a rational root is found, repeat the process on the quotient obtained from the bottom



hto n Mifflin

Use the Rational Root Theorem to identify possible rational roots. Check each by using synthetic

= x 3 + 4x2 + x - 6 Each of the z How are the zeros of ƒ( x) related to the standard form of the function? polynomial function is a factor of the constant term in the standard form

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

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



© Ho ug

COMMON CORE

Date

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

Common Core Math Standards COMMON CORE

Class



RDCOVER PA Turn to thes ind this less hardcover st edition.

Reflect

EXPLORE

In general, how are the zeros of a polynomial function related to the function written in standard form? Each of the numerators of the zeros is a factor of the constant term. Each of the

1.

Relating Zeros and Coeffic Polynomial Functions

denominators of the zeros is a factor of the leading coefficient. Discussion Does the relationship from the first Reflect question hold if the zeros are all integers? Explain. Yes; If the zeros are all integers, each of them can be written with a denominator of 1. Each

2.

of the numerators is still a factor of the constant term.

(

)(

)(

INTEGRATE TECHNOLOGY

)

3 5 1 __ __ If you use the zeros, you can write the factored form of g( x) as g( x) = x + __ 2 x - 4 x - 6 , 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. In each factor, the denominator of the fraction becomes the coefficient of the variable.

3.

Students have the option of complet activity either in the book or online.

b In general, if the zero is -_ a, the factor can be written as(ax + b ).

Explain 1

QUESTIONING STRATEGIE

Finding Zeros Using the Rational Zero Theorem

What is the relationship betw a polynomial function and th function? The zeros are the values o setting each factor equal to 0 and so

If a polynomial function p( x) is equal to ( a 1x + b1)( a 2x + b2)( a 3x + b3) , where a1, a 2 , a3, b1 , b 2 , and b3 are integers, the leading coefficient of p(x )will be the product a1a2 a3 and the constant term will be b

b

b

3 the product b1b 2b 3. The zeros of p(x) will be the rational numbers -a__11 , -a__22 , -__ a 3.

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.

If a zero of a polynomial func do you know about the coeffi polynomial is written in standard fo of the constant term and 13 is a fact coefficient.

Rational Zero Theorem

(

)

m is a zero of p( x) p( m __) = 0 , If p( x ) is a polynomial function with integer coefficients, and if _ n n then m is a factor of the constant term of p( x) and n is a factor of the leading coefficient of p(x).



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 + 2x2 - 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.

Module 7

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Example 1

EXPLAIN 1 Finding Zeros Using the R Theorem

m _ n

1

2

-19

-20

1

1

3

-16

-36

2

1

4

-11

-42

QUESTIONING STRATEGIE

4

1

6

5

0

5

1

7

16

60

Is every zero of a polynomial represented in the set of numb Rational Zero Theorem? No. The Ra Theorem gives only those zeros that numbers. A polynomial function can that are irrational numbers or imagi

342

Lesson 1

PROFESSIONAL DEVELOPMENT 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

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

AVOID COMMON ERRORS

( x - 4) (x 2 + 6x + 5) = 0

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.

(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 - 4x3 - 7x 2 + 22x + 24 a. Use the Rational Zero Theorem to find all possible rational zeros. Factors of 24: ± 1

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

,± 2

,± 3

,± 4

,± 6

,± 8

, ± 12 , ±

b. Test the possible zeros. Use a synthetic division table.

_m

1

-4

-7

22

24

1

1

-3

-10

12

36

2

1

-2

-11

0

24

1

-1

-10

-8

0

n

3

c. Factor the polynomial. The synthetic division by 3 results in a remainder of 0 so 3 is a zero and the polynomial in factored form is given as follows: (x -

3

10 x -

) (x 3 - x2 -

8

)=0

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

10 x -

8.

Factors of -8: ± 1 , ± 2 , ± 4 , ± 8

© Houghton Miff lin Harcourt Publishing Company

e. Test the possible zeros. Use a synthetic division table.

_m

1

-1

-10

-8

1

1

0

-10

-18

2

1

1

-8

-24

4

1

3

2

0

n

f. 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: (x (x -

x=

3 3

)(x )(x -

3 ,x=

4 4

)(

1

)( x +

4, x =

x2 +

2

3 x+ 1

)(x +

-2 , or x =

2

)= 0

)=0

-1

The zeros are x = 3, x = 4, x = -2, and x = -1.

Module 7

343

COLLABORATIVE LEARNING Small Group Activity Have students work in groups of 3–4 students. Instruct each group fifth-degree polynomial function with rational zeros, not all of whi Ask them to write their functions in standard from. Have groups ex functions, and have each group create a poster showing how to app

Reflect

4.

INTEGRATE MATHEMATIC PRACTICES Focus on Math Connection MP.1 Remind students that a zero

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? To find the zeros of a 4th degree polynomial using synthetic division, you need to use synthetic

division to reduce that polynomial to a 3rd degree polynomial and then use synthetic division again to reduce that polynomial to a quadratic polynomial that can be factored, if possible. 5.

number from the domain that the fu 0. Discuss that, for this reason, a gra function will have an x-intercept at e Students can then make a concrete c between the rational zeros they iden function, and the role the zeros play the function.

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? 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. 6.

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

()

from the synthetic division array for ƒ__12 is 2 x2 + 2x - 12. Show how to write the factored form of ƒ( x ) = 2 x 3 + x 2 - 13x + 6 using integer coefficients. 1 as a zero and the quotient 2x2 + 2x - 12 you can write f( x) = 2x 3 + x2 - 13x + 6 Using __ 2 1 as f( x) = x - _ ( 2x2 + 2x - 12 ) . 2

( ) 1 ( 2) x f( x) = (x - _1)( 2x + 2x - 12 ) = (x - _ ( + x - 6) 2) 2 2

2

= (2x - 1 )(x2 + x - 6 ) = (2x - 1 ) (x + 3) (x - 2 ) Your Turn

7.

Find the zeros of ƒ( x ) = x 3 + 3x2 - 13x- 15.

a. Use the Rational Zero Theorem. Factors of -15: ±1, ±3, ±5, ±15 m _

1

-3

-13

-15

1

1

4

-9

-24

3

1

6

5

0

n

© Houghton Miff lin Harcourt Publishing Company

b. Test the possible zeros to find one that is actually a zero.

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

Module 7

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

344

Lesson 1

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

Explain 2

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.

Solving a Real-World Problem Using the Rational Root Theorem

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

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.

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.

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.

A. Analyze Information

What information is obtained by applying the Rational Zero Theorem to a polynomial function? A list of all possible rational zeros of the function

and the box must have a volume of

1 The important information is that the base width must be inch shorter than the base length, the height must be 3 inches taller than 3 times the base length, 6 cubic inches.

B. Formulate a Plan

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Write an equation to model the volume of the box. Let x represent the base length in inches. The base width is x - 1 and the 3( x + 1. ) height is 3x + 3 , or 1 _ ℓw h = V 3 1 ( x ) (x - 1 ) (3)(x + 1 ) = 6 3 1 x 3 - 1x 6 0 =

_

Module 7

345

C. Solve

INTEGRATE MATHEMATIC PRACTICES Focus on Critical Thinking MP.3 Prompt students to recogniz

Use the Rational Root Theorem to find all possible rational roots. Factors of -6: ± 1 , ± 2 , ± 3 , ± 6 Test the possible roots. Use a synthetic division table. m _

1

0

-1

-6

1

1

1

0

-6

2

1

2

3

0

3

1

3

8

18

n

Factor the polynomial. The synthetic division by so

(

2

rational roots found by factoring the quadratic polynomial must be numb identified as possible rational roots i help them to catch errors in factorin performing the synthetic division.

results in a remainder of 0,

2 is a root and the polynomial in factored form is as follows: 1 x - 2 ) ( 1 x2 + 2 x+ 3 ) =0

The quadratic polynomial produces only complex roots, so the only possible answer for the base length is height is

2 inches. The base width is

1 inch and the

9 inches.

D. Justify and Evaluate The x-coordinates of...