LESSON
7.1
Name
Finding Rational
Solutions of
Polynomial Equations
Class
7.1
Explore
The student is expected to:
A-APR.2
Mathematical Practices
Relating Zeros and Coefficients of
Polynomial Functions
A
Identify the zeros of the polynomial function. The zeros are x = -2, x = 1, and x = -3.
B
Multiply the factors to write the function in standard form.
MP.2 Reasoning
f(x) = (x + 2)(x - 1)(x + 3)
= (x 2 + 2x - x - 2)(x + 3)
= (x 2 + x - 2)(x + 3)
Language Objective
Explain to a partner how to identify the factors of a polynomial function.
= x 3 + 3x 2 + x 2 + 3x - 2x - 6
C
ENGAGE
PREVIEW: LESSON
PERFORMANCE TASK
= x 3 + 4x 2 + x - 6
How are the zeros of ƒ(x) related to the standard form of the function? Each of the zeros of the
polynomial function is a factor of the constant term in the standard form.
© Houghton Mifflin Harcourt Publishing Company
D
Now consider the polynomial function g(x) = (2x + 3)(4x - 5)(6x - 1). Identify the zeros
of this function.
5
1
3
The zeros are x = -_
,x=_
, and x = _
.
4
6
2
E
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 - 8x 2 + 12x 2 - 2x - 90x + 15
F
= 48x 3 + 4x 2 - 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, and each of the
denominators is a factor of the leading coefficient, 48.
Module 7
be
ges must
EDIT--Chan
DO NOT Key=NL-A;CA-A
Correction
Lesson 1
341
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Date
Class
7.1
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and a numbe ). Also
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is p(a), so
A-APR.2 Know
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.3
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A-APR.3, A-CED
Zeros and ns
ting
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. Consider
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Explore
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Polynom
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coefficients
n and the
x = -3.
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)(x + 3).
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of a polyno ) = (x + 2)(x -1
are x = -2,
The zeros
ƒ(x
function
The zeros
polynomial
ion: How
Identify the

do you find
zeros of the
the ration
Resource
Locker
HARDCOVER PAGES 247254
Turn to these pages to
find this lesson in the
hardcover student
edition.
function.
polynomial
rd form.
on in standa
the functi
s to write
the factor
)
Multiply
1)(x + 3
+ 2)(x f(x) = (x
)(x + 3)
2
x-x-2
= (x + 2
3)
2
- 2)(x +
= (x + x
6
2
3x - 2x 3
x2 + x +
=x +3

the
zeros of
Each of the
on?
the functi
6
rd form of
3
x2 + x ard form.
to the standa
=x +4
in the stand
of ƒ(x) related
ant term
the zeros
r of the const
zeros
is a facto
Identify the
l function
)(6x - 1).
3)(4x - 5
+
2x
(
on g(x) =
omial functi
er the polyn
Now consid
_1 .
on.
_3 , x = _54, and x = 6
of this functi
are x = - 2
The zeros
rd form.
on in standa
the functi
s to write
the factor
1)
Multiply
5)(6x 3)(4x +
2x
(
=
- 1)
g(x)
- 15)(6x
2
10x + 12x
= (8x )
1
(6x
2
2x - 15)
= (8x +
x + 15
2
- 2x - 90
2
x
12
+
3
8x
on?
= 48x of the
the functi
+ 15
2
rd form of
3
15, and each
4x - 92x
to the standa
ant term,
= 48x +
of g(x) related
r of the const
the zeros
is a facto
How are
s of the zeros
icient, 48.
numerator
ng coeff
Lesson 1
Each of the
the leadi
factor of
tors is a
denomina
How are
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polynomia
y
g Compan

Publishin
A2_MNLESE385894_U3M07L1.indd 341
s
ion
tional Solut
Finding Ra ial Equations
of Polynom
Name
Quest
Essential
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.
© Houghto
n Mifflin
Harcour t


341
Module 7
Lesson 7.1
L1.indd
4_U3M07
SE38589
A2_MNLE
341
Resource
Locker
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).
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
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.
Finding Rational Solutions
of Polynomial Equations
Essential Question: How do you find the rational roots of a polynomial equation?
Common Core Math Standards
Essential Question: How do you find
the rational roots of a polynomial
equation?
Date
341
3/19/14
2:39 PM
3/19/14 2:37 PM
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 Coefficients of
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
)
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.
In each factor, the denominator of the fraction becomes the coefficient of the variable.
3.
Students have the option of completing the Explore
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 STRATEGIES
Finding Zeros Using the Rational Zero Theorem
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 polynomial function p(x) is equal to (a 1x + b 1)(a 2x + b 2)(a 3x + 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 1a 2 a 3 and the constant term will be
b
b
b
1
__2
__3
the product b 1b 2b 3. The zeros of p(x) will be the rational numbers -__
a1 , - a2 , - a3 .
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
(
7 , what
If a zero of a polynomial function is ___
13
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.
)
m is a zero of p(x) p __
If p(x) is a polynomial function with integer coefficients, and if _
(mn ) = 0 ,
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 + 2x 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.
Module 7
m
_
1
2
-19
-20
1
1
3
-16
-36
2
1
4
-11
-42
4
1
6
5
0
5
1
7
16
60
n
342
© Houghton Mifflin Harcourt Publishing Company
Example 1
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.
Lesson 1
PROFESSIONAL DEVELOPMENT
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
and constant term, establishing the Rational Zero Theorem.
Finding Rational Solutions of Polynomial Equations
342
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:
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 - 4x 3 - 7x 2 + 22x + 24
a. Use the Rational Zero Theorem to find all possible rational zeros.
QUESTIONING STRATEGIES
Factors of 24: ± 1 , ±
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 , ± 24
b. Test the possible zeros. Use a synthetic division table.
m
_
1
-4
-7
22
1
1
-3
-10
12
36
2
1
-2
-11
0
24
3
1
-1
-10
-8
0
n
24
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
)(x 3 - x 2 -
10 x -
8
)=0
d. Use the Rational Zero Theorem again to find all possible rational zeros of
g(x) = x 3 - x 2 -
10
8
x-
.
Factors of -8: ± 1 , ± 2 , ± 4 , ± 8
© Houghton Mifflin 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 -
3
(x -
3
x=
3
)(
)(x -
4
)(x -
4 )(x +
,x=
4
1
x2 +
2
3
)(x +
x+
1
, x = -2 , or x = -1
2
)=0
)=0
The zeros are x = 3, x = 4, x = -2, and x = -1.
Module 7
343
Lesson 1
COLLABORATIVE LEARNING
A2_MNLESE385894_U3M07L1.indd 343
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.
343
Lesson 7.1
7/7/14 10:24 AM
Reflect
4.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Remind students that a zero of a function is a
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 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.
5.
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.
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.
Using synthetic division, you find that __12 is a zero of ƒ(x) = 2x 3 + x 2 - 13x + 6. The quotient
()
from the synthetic division array for ƒ __12 is 2x 2 + 2x - 12. Show how to write the factored form of
ƒ(x) = 2x3 + x 2 - 13x + 6 using integer coefficients.
1
as a zero and the quotient 2x 2 + 2x - 12 you can write f(x) = 2x 3 + x 2 - 13x + 6
Using __
2
1 (2x 2 + 2x - 12).
as f(x) = x - _
2
( )
1 (2x + 2x - 12) = x - _
f(x) = (x - _
( 12 )(2)(x + x - 6)
2)
2
2
= (2x - 1)(x 2 + x - 6) = (2x - 1)(x + 3)(x - 2)
Your Turn
7.
Find the zeros of ƒ(x) = x 3 + 3x 2 - 13x- 15.
a. Use the Rational Zero Theorem. Factors of -15: ±1, ±3, ±5, ±15
m
_
n
1
-3
-13
-15
1
1
4
-9
-24
3
1
6
5
0
© Houghton Mifflin 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)(x 2 + 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
A2_MNLESE385894_U3M07L1.indd 344
10/16/14 11:22 PM
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.
Finding Rational Solutions of Polynomial Equations
344
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
m
the polynomial equation p(x) = 0 can be written in the form __
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).
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,
cubic inches.
Write an equation to model the volume of the box.
© Houghton Mifflin Harcourt Publishing Company
Let x represent the base length in inches. The base width is
height is 3x + 3 , or 3(x + 1) .
1 ℓw h = V
_
3
1 ( x )(x - 1 )(3) x + 1 = 6
(
)
3
1 x3 - 1 x - 6 = 0
x-1
and the
_
A2_MNLESE385894_U3M07L1 345
Lesson 7.1
6
B. Formulate a Plan
Module 7
345
in
tory g
His arkin
m
the
345
Lesson 1
6/28/14 2:13 PM
C. Solve
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Prompt students to recognize that any
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
2
so
(
1
2
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.
results in a remainder of 0,
is a root and the polynomial in factored form is as follows:
2 )( 1
xx2 + 2 x + 3 ) = 0
The quadratic polynomial produces only complex roots, so the only possible
2
answer for the base length is
height is
9
inches. The base width is
1
inch and the
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 (2, 6) .
Since the x-coordinate is
2
, the answer is correct.
Your Turn
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?
A.
B.
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.
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
© Houghton Mifflin Harcourt Publishing Company
8.
x 3 + x 2 - 6x - 56 = 0
Module 7
A2_MNLESE385894_U3M07L1 346
346
Lesson 1
6/27/14 10:32 PM
Finding Rational Solutions of Polynomial Equations
346
C.
ELABORATE
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 Have students discuss how they could use a
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)(x 2 + 5x + 14) = 0
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.
D.
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.
Using a graphing calculator, the graphs intersect at (4, 56), which validates the answer.
Elaborate
9.
For a polynomial function with integer coefficients, how are the function’s coefficients and rational zeros
related?
m
The rational zeros of a polynomial function with integer coefficients are in the form __
n,
where m is a factor of the constant term and n is a factor of the leading coefficient.
QUESTIONING STRATEGIES
10. Describe the process for finding the rational zeros of a polynomial function with integer coefficients.
Using the Rational Zero Theorem to find all possible rational zeros, test the possible zeros
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.
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
p
the form __
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.
and factor the polynomial.
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?
The theorem is useful in this case because it allows you to find the rational roots of the
© Houghton Mifflin Harcourt Publishing Company
SUMMARIZE THE LESSON
to find one that is actually a zero by using a synthetic division table to organize the work
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.
12. Essential Question Check-In What does the Rational Root Theorem find?
The Rational Root Theorem finds the possible rational roots of a polynomial equation.
Module 7
347
Lesson 1
LANGUAGE SUPPORT
A2_MNLESE385894_U3M07L1 347
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.
347
Lesson 7.1
6/27/14 10:32 PM
EVALUATE
Evaluate: Homework and Practice
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.
Factors of −8 : ±1, ±2, ±4, ±8
5 is a zero.
(x - 4)(x 2 + 3x + 2) = 0
(x - 4)(x + 2)(x + 1) = 0
x = 5, x = -3, or x = -4
f(x) = (x - 4)(x + 2)(x + 1)
f(x) = (x - 5)(x + 3)(x + 4)
j(x) = 2x 3 - x 2 - 13x - 6
4.
Factors of −6 : ±1, ±2, ±3, ±6
g(x) = x 3 - 9x 2 + 23x − 15
Factors of −15 : ±1, ±3, ±5, ±15
3 is a zero.
1 is a zero.
(x - 3)(2x 2 + 5x + 2) = 0
(x - 1)(x 2 - 8x + 15) = 0
(x - 1)(x -5)(x - 3) = 0
(x - 3)(2x + 1)(x + 2) = 0
x = 1, x = 5, or x = 3
1
x = 3, x = -_
, or x = -2
g(x) = (x - 1)(x - 5)(x - 3)
2
j(x) = (x - 3)(2x + 1)(x + 2)
h(x) = x 3 - 5x 2 + 2x + 8
6.
h(x) = 6x 3 - 7x - 9x − 2
(x - 2)(6x 2 + 5x + 1) = 0
1
1
x = 2, x = -_
, or x = -_
2
3
h(x) = (x - 2)(x - 4)(x + 1)
h(x) = (x - 2)(2x + 1)(3x + 1)
s(x) = x 3 - x 2 − x + 1
8.
t(x) = x 3 + x 2 − 8x − 12
Factors of 1 : ±1
Factors of −12 : ±1, ±2, ±3, ±4, ±6, ±12
1 is a zero.
3 is a zero.
(x - 1)(x - 1) = 0
(x - 1)(x + 1)(x - 1) = 0
(x - 3)(x 2 + 4x + 4) = 0
(x - 3)(x + 2)(x + 2) = 0
s(x) = (x - 1)(x + 1)(x - 1)
t(x) = (x - 3)(x + 2)(x + 2)
2
x = 1 or x = -1
Module 7
Exercise
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
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.
x = 3 or x = -2
Lesson 1
348
A2_MNLESE385894_U3M07L1 348
© Houghton Mifflin Harcourt Publishing Company
(x - 2)(2x + 1)(3x + 1) = 0
x = 2, x = 4, or x = -1
Practice
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Students can use patterns in the signs of the
2 is a zero.
(x - 2)(x 2 - 3x - 4) = 0
(x - 2)(x - 4)(x + 1) = 0
Concepts and Skills
2
Factors of −2 : ±1, ±2
Factors of 8 : ±1, ±2, ±4, ±8
2 is a zero.
7.
ASSIGNMENT GUIDE
(x - 5)(x 2 + 7x + 12) = 0
(x - 5)(x + 3)(x + 4) = 0
x = 4, x = -2, or x = -1
5.
ƒ(x) = x 3 + 2x 2- 23x - 60
Factors of −60 : ±1, ±2, ±3, ±4, ±5, ±6,
±10, ±12, ±15, ±20, ±30, ±60
4 is a zero.
3.
• Online Homework
• Hints and Help
• Extra Practice
Depth of Knowledge (D.O.K.)
Mathematical Practices
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
16/10/14 10:37 AM
Finding Rational Solutions of Polynomial Equations
348
9.
AVOID COMMON ERRORS
k(x) = x 4 + 5x 3 - x 2 − 17x + 12
10. g(x) = x 4 - 6x 3 + 11x 2 - 6x
g(x) = x(x 3 - 6x 2 + 11x - 6)
Factors of 12 : ±1, ±2, ±3, ±4, ±6, ±12
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.
Factors of -6 : ±1, ±2, ±3, ±6
1 is a zero.
1 is a zero.
Factor the polynomial.
(x − 1)(x 3 + 6x 2 + 5x − 12)
(x)(x - 1)(x 2 - 5x + 6) = 0
1 is a zero.
x = 1, x = 0, x = 3, or x = 2
Factors of 12 : ±1, ±2, ±3, ±4, ±6, ±12
(x)(x - 1)(x - 3)(x - 2) = 0
g(x) = (x)(x - 1)(x - 3)(x - 2)
(x - 1)(x - 1)(x 2 + 7x + 12) = 0
(x - 1)(x - 1)(x + 3)(x + 4) = 0
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).
x = 1, x = -3, or x = -4
k(x) = (x - 1)(x - 1)(x + 3)(x + 4)
11. h(x) = x 4 - 2x 3 - 3x 2 + 4x + 4
12. ƒ(x) = x 4 - 5x 2 + 4
Factors of 4 : ±1, ±2, ±4
Factors of 4 : ±1, ±2, ±4
2 is a zero.
1 is a zero.
(x − 2)(x − 3x − 2)
f(x) = (x - 1)(x 3 + x 2 − 4x − 4)
2 is a zero.
2 is a zero.
3
Factors of −4 : ±1, ±2, ±4
Factors of -2 : ±1, ±2
(x - 2)(x - 2)(x + 2x + 1) = 0
(x - 2)(x - 2)(x + 1)(x + 1) = 0
(x - 1)(x - 2)(x 2 + 3x + 2) = 0
(x - 1)(x - 2)(x + 2)(x + 1) = 0
2
x = -1 or x = 2
x = 1, x = 2, x = -2, or x = -1
f(x) = (x - 1)(x - 2)(x + 2)(x + 1)
© Houghton Mifflin Harcourt Publishing Company
h(x) = (x - 2)(x - 2)(x + 1)(x + 1)
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?
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
2x 3 + 5x 2 - 18x = 45
2x + 5x 2 - 18x - 45 = 0
3
Factors of -45: ±1, ±3, ±5, ±9, ±15, ±45
3 is a root.
(x - 3)(2x 2 + 11x + 15) = 0, so (x - 3)(2x + 5)(x + 3) = 0
-5
x = 3, x = ___
, or x = -3
2
Length cannot be negative. The length must be is 3 inches. The width is 1 inch and
the height is 15 inches.
Module 7
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Lesson 1
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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?
CRITICAL THINKING
Discuss with students why the Rational Root
Theorem works, by applying it to a quadratic
equation, such as 2x 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.
Let x represent the side of the square base in yards. The height is x + 2.
_1 ℓwh = V
3
_1(x)(x)(x + 2) = 147
3
_1(x 3 + 2x 2) = 147
3
x 3 + 2x 2 = 441
x 3 + 2x 2 - 441 = 0
7 is a root.
Factors of -441: ±1, ±3, ±7, ±9, ±21, ±49, ±63, ±147, ±441
(x - 7)(x 2 + 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.
15. Engineering A paper company is designing a new, pyramidshaped 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 height in inches. The length is
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©James
Kingman/Shutterstock
x - 4 and the width is x - 3.
_1 ℓwh = V
3
1
_(x - 4)(x - 3)(x) = 12
3
_1(x 3 - 7x 2 + 12x) = 12
3
x 3 - 7x 2 + 12x = 36
x 3 - 7x 2 + 12x - 36 = 0
Factors of -36: ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36
6 is a root.
(x - 6)(x 2 - 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.
Module 7
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Lesson 1
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Finding Rational Solutions of Polynomial Equations
350
16. Match each set of roots with its polynomial function.
B
3
A. x = 2, x = 3, x = 4
ƒ(x) = (x + 2)(x + 4) x – _
2
5 x+_
7
3
1 x–_
C
ƒ(x) = x – _
B. x = –2, x = –4, x = _
4
2
2
3
5 , x = – _
7
1, x = _
A
ƒ(x) = (x – 2)(x – 3)(x – 4)
C. x = _
4
2
3
D ƒ(x) = x + _
6, x = 4
6 (x – 4)
4 x–_
4, x = _
D. x = – _
7
5
7
5
PEER-TO-PEER DISCUSSION
Ask students to discuss with a partner how the
Rational Root Theorem, in conjunction with the Zero
Product Property, enables them to solve real-world
problems that can be modeled by polynomial
equations. The Rational Root Theorem can be used
to identify possible solutions. Identifying one or
more of the solutions from the list of possible
solutions can help you to write the equation in
factored form. You can then use the Zero Product
Property to set each factor equal to zero and solve
for other possible solutions.
(
(
)(
)(
(
)(
)
)
)
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.
The zeros of f(x) are x = -3, x = 4, and x = 3.
f(x) = (x + 3)(x - 4)(x - 3) = (x 2 + 3x - 4x - 12)(x - 3)
= (x 2 - x - 12)(x - 3) = x 3 - 3x 2 - x 2 + 3x - 12x + 36
= x 3 - 4x 2 - 9x + 36
The zeros of f(x) are all factors of the constant term in the polynomial function.
H.O.T. Focus on Higher Order Thinking
18. Critical Thinking Consider the polynomial function g(x) = 2x 3 - 6x 2 + πx + 5. Is it
possible to use the Rational Zero Theorem and synthetic division to factor this polynomial?
Explain.
JOURNAL
Have students describe how they could use the
Rational Zero Theorem to write a polynomial
function in intercept form.
No; it is not possible because the function contains a term, πx, whose
coefficient is irrational and, therefore, not an integer.
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.
b
For any factor (ax + b), a zero occurs at -_
a . Sabrina forgot to include the
© Houghton Mifflin Harcourt Publishing Company
negative sign when converting from her factors to the zeros.
c is a rational zero of a polynomial function p(x), explain why
20. Justify Reasoning If _
b
bx - c must be a factor of the polynomial.
()
c
c
1(
b
p(x) = (x - _
q(x) and p(x) = _
x-_
q(x) = _
bx - c)q(x), which
b)
b)
b
b(
Since p _c = 0, x - _c is a factor of p(x) by the Factor Theorem. So,
b
b
shows that bx - c is a factor of p(x).
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.
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.
Module 7
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Lesson 7.1
351
Lesson 1
16/10/14 10:53 AM
Lesson Performance Task
AVOID COMMON ERRORS
For the years from 2001–2010, the number of Americans traveling to other countries by plane
can be represented by the polynomial function A(t) = 20t 4 - 428t 3 + 2760t 2 - 4320t + 33,600 ,
where A is the number of thousands of Americans traveling abroad
by airplane and t is the number of years since 2001. In which year
were there 40,000,000 Americans traveling abroad? Use the
Rational Root Theorem to find your answer.
[Hint: consider the function’s domain and range before finding all
possible rational roots.]
Some students may set A(t) equal to 40,000,000,
which is the number given in the problem. Ask
students to check the units of A. thousands of
Americans. Have students divide 40,000,000 by 1,000
to get the correct value for A, 40,000. More precisely,
A is 40,000 thousands of Americans.
A(t) = 20t 4 - 428t 3 + 2760t 2 - 4320t + 33,600
40,000 = 20t 4 - 428t 3 + 2760t 2 - 4320t + 33,600
0 = 20t 4 - 428t 3 + 2760t 2 - 4320t - 6400
QUESTIONING STRATEGIES
Factors of -6400 between 0 and 9: 1, 2, 4, 5, 8. Test the possible roots:
m
_
n
20
-428
2760
-4320
-6400
1
20
-408
2352
-1968
-8368
2
20
-388
1984
-352
-7104
4
20
-348
1368
1152
-1792
5
20
-328
1120
1280
0
-268
616
608
-1536
8
20
(x - 5)(20x - 328x + 1120x + 1280)
3
Why is it useful to know a function’s domain
when solving for the roots? If the domain
consists only of rational numbers, then the roots
must be rational. For example, if the domain
consists of the integers from 0 to 9, then the roots
must be rational because integers are rational
numbers.
2
Why does the domain consist only of
integers? The domain is the number of years
since 2001. The function only makes sense for
integer values.
Factors of 1280 between 0 and 9: 1, 2, 4, 5, 8. Test the possible roots:
_
20
-328
1120
1280
1
20
-308
812
2092
2
20
-288
544
2368
4
20
-248
128
1792
5
20
-228
-20
1180
8
20
-168
-224
-512
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Paul
Seheult/Eye Ubiquitous/Corbis
m
n
Because the cubic polynomial factor has no rational roots between 0 and 9, x = 5 years
returns the only solution. In other words, there were 40,000,000 Americans traveling
overseas by air in 2006.
Module 7
352
Lesson 1
EXTENSION ACTIVITY
A2_MNLESE385894_U3M07L1 352
Have students research the factors that affect tourist numbers, such as changes in
economic status, or the safety of a destination. Have students discuss who might
use a model of tourist numbers like A(t) and how it might be used. Ask students
to describe situations in which it would be useful to input a value of t to calculate
the number of tourists, and in what situations it would be useful to do the
inverse—use a given number of tourists and solve for the roots.
6/9/15 12:25 AM
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.
Finding Rational Solutions of Polynomial Equations
352