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LESSON
7.2
Name
Finding Complex
Solutions of
Polynomial Equations
Class
7.2
Finding Complex Solutions
of Polynomial Equations
Essential Question: What do the Fundamental Theorem of Algebra and its corollary tell you
about the roots of the polynomial equation p(x) = 0 where p(x) has
degree n?
Common Core Math Standards
A-APR.2
You have used various algebraic and graphical methods to find the roots of a polynomial
equation p(x) = 0 or the zeros of a polynomial function p(x). Because a polynomial can have a
factor that repeats, a zero or a root can occur multiple times.
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 N-CN.9(+), A-REI.11, A-APR.3,
F-IF.7c
The polynomial p(x) = x 3 + 8x 2 + 21x + 18 = (x + 2)(x + 3) has -2 as a zero once
and -3 as a zero twice, or with multiplicity 2. The multiplicity of a zero of p(x) or a root of
p(x) = 0 is the number of times that the related factor occurs in the factorization.
2
Mathematical Practices
In this Explore, you will use algebraic methods to investigate the relationship between the
degree of a polynomial function and the number of zeros that it has.
MP.7 Using Structure
Language Objective
A
Find all zeros of p(x) = x 3 + 7x 2. Include any multiplicities greater than 1.
p(x) = x 3 + 7x 2
Complete a “Solving Polynomial Equations” chart with a partner.
Factor out the GCF.
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo
and what variables you might use to describe the
amount of violence in a movie. Then preview the
Lesson Performance Task.
p(x) = x 2 (x + 7)
)
(
What are all the zeros of p(x)? 0 mult. 2 , -7
© Houghton Mifflin Harcourt Publishing Company
ENGAGE
The equation has exactly n complex roots provided
that you count the multiplicities of the roots.
B
Find all zeros of p(x) = x 3 - 64. Include any multiplicities greater than 1.
p(x) = x 3 - 64
p(x) = x - 4 x 2 + 4x + 16
What are the real zeros of p(x)?
4
)
Solve x + 4x + 16 = 0 using the quadratic formula.
2
―――
-b ± √b - 4ac
x = __
2a
2
―――
――――――
√
√
_
-4 ± 4 - 4 ∙ 1 ∙ 16
-4 ± -48
-4 ± 4i √3
x = ___ x = _x = __
2
2
2∙ 1
2
―
x = -2 ± 2i √3
―
―
√
√
What are the non-real zeros of p(x)? -2 + 2i 3 , -2 -2i 3
Module 7
Lesson 2
353
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.3, F-IF.7c
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A2_MNLESE385894_U3M07L2.indd 353
HARDCOVER PAGES 255264
g the
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InvestigatinPolynomial Functio
mial
a
of a polyno
a
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Zeros of
can have
Explore
Turn to these pages to
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hardcover student
edition.
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in
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Find all zeros
What are
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
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Find all zeros
Publishin
Factor the
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What are
of p(x)?
difference
the real zeros
2
4x + 16
Solve x +
n Mifflin
0 (mult.
2), -7
r than 1.
licities greate
e any multip
x3 - 64. Includ
3
- 64
p(x) = x
16
2 + 4x +
- 4 x
p(x) = x
of two cubes.
of p(x) =
atic formu
the quadr
2
© Houghto
x=
±
-b
_
2a
L2.indd
――――――
√
What are
4_U3M07
)
la.
―――
_
√3
-4 ± 4i
__
-4 ± -48 x =
2
∙ 16
2
4 -4∙ 1
=_
2
-4 ± __ x
_
2∙ 1
Module 7
SE38589
)(
(
4
of p(x)?
= 0 using
―
――
- 4ac
√b_
x = -2 ±
A2_MNLE
Includ
x3 + 7x .
2
3
+ 7x
p(x) = x
2 (x + 7)
p(x) = x
all the zeros
x=
Lesson 7.2
)(
(
Factor the difference of two cubes.

353
Resource
Locker
Investigating the Number of Complex
Zeros of a Polynomial Function
Explore
The student is expected to:
Essential Question: What do the
Fundamental Theorem of Algebra and
its corollary tell you about the roots of
the polynomial equation p(x) = 0
where p(x) has degree n?
Date
353
―
2i √3
eal zeros
the non-r
of p(x)?
√
-2 + 2i
―
√3 , -2 -2i
―
√3
Lesson 2
353
3/19/14
3:24 PM
3/19/14 3:23 PM
C
Find all zeros of p(x) = x 4 + 3x 3 - 4x 2 - 12x. Include any multiplicities greater than 1.
p(x) = x 4 + 3x 3 - 4x 2 - 12x
Factor out the GCF.
Group terms to begin
factoring by grouping.
(
p(x) = x x 3 + 3x 2 - 4x - 12
(
(
p(x) = x (x 3 + 3x 2) - 4x + 12
(
))
Factor out common monomials.
p(x) = x x 2 (x + 3) - 4 (x + 3)
Factor out the common binomial.
p(x) = x(x + 3)(x 2 - 4)
Factor the difference of squares.
p(x) = x(x + 3)
What are all the zeros of p(x)? 0, -3, -2, 2
D
)
(
x+2
)(
EXPLORE
Investigating the Number of Complex
Zeros of a Polynomial Function
)
x-2
INTEGRATE TECHNOLOGY
Students have the option of completing the Explore
activity either in the book or online.
)
QUESTIONING STRATEGIES
Find all zeros of p(x) = x 4 - 16. Include any multiplicities greater than 1.
p(x) = x 4 - 16
Factor the difference of squares.
Factor the difference of squares.
(
p x =(
p(x) =
( )
x2 - 4
x+2
What are the real zeros of p(x)? -2, 2
When would you need to use the quadratic
formula to find a zero? When one of the
factors of the polynomial is a non-factorable
quadratic polynomial.
) (x + 4)
)( x - 2 )(x + 4)
2
2
Solve x 2 + 4 = 0 by taking square roots.
x2 + 4 = 0
x 2 = -4
_
x = ±√ -4
© Houghton Mifflin Harcourt Publishing Company
x = ± 2i
What are the non-real zeros of p(x)? -2i, 2i
Module 7
354
Lesson 2
PROFESSIONAL DEVELOPMENT
Learning Progressions
A2_MNLESE385894_U3M07L2.indd 354
3/19/14 3:23 PM
Students have learned factoring techniques in earlier lessons, and a more general
technique for finding zeros of polynomial functions and solutions of polynomial
equations based on the Rational Zero/Root Theorem in the previous lesson. They
have also learned how to use the quadratic formula to solve quadratic equations.
In this lesson, students pull all these techniques together in order to understand
and use the Fundamental Theorem of Algebra.
Finding Complex Solutions of Polynomial Equations
354
E
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Encourage students to look for patterns in
Find all zeros of p(x) = x 4 + 5x 3 + 6x 2 -4x -8. Include multiplicities greater than 1.
By the Rational Zero Theorem, possible rational zeros are ±1, ±2, ±4, and ±8.
Use a synthetic division table to test possible zeros.
their results. They can make connections between the
degree of each polynomial and the number of zeros,
and between a function’s characteristics and their
effects on the nature of its zeros. Students can also be
prompted to make conjectures about the number of
each type of zero (real and non-real) that could exist
for polynomials of varying degrees.
m
_
n
1
5
6
-4
-8
1
1
6
12
8
0
The remainder is 0, so 1 is/is not a zero.
(
)
p(x) factors as (x - 1) x 3 + 6x 2 + 12x + 8 .
Test for zeros in the cubic polynomial.
m
_
n
1
6
12
8
1
1
7
19
27
-1
1
5
7
1
2
1
8
28
64
-2
1
4
4
0
-2 a zero.
(
)
p(x) factors as (x - 1)(x + 2) x 2 + 4x + 4 . The quadratic is a perfect square
trinomial.
(x + 2)3 .
1, -2 (mult. 3)
What are all the zeros of p(x)?
So, p(x) factors completely as p(x) = (x - 1)
© Houghton Mifflin Harcourt Publishing Company
F
Complete the table to summarize your results from Steps A–E.
Polynomial Function in
Standard Form
Real Zeros
and Their
Multiplicities
Polynomial Function
Factored over the Integers
Non-real Zeros
and Their
Multiplicities
p(x) = x 3 + 7x 2
p(x) = x 2 (x + 7)
0 (mult. 2); -7
p(x) = x 3 - 64
p(x) = (x - 4)(x 2 + 4x + 16)
4
p(x) = x 4 + 3x 3 - 4x 2 - 12x
p(x) = x(x + 3)(x + 2)(x - 2) 0; -3; -2; 2
None
p(x) = x - 16
p(x) = (x - 2)(x + 2)(x + 4)
-2; 2
-2i; 2i
p(x) = x 4 + 5x 3 + 6x 2 - 4x - 8
p(x) = (x - 1)(x + 2)
1, -2 (mult. 3)
None
4
2
Module 7
3
355
None
―
-2 - 2i √―
3
-2 + 2i √3 ;
Lesson 2
COLLABORATIVE LEARNING
A2_MNLESE385894_U3M07L2.indd 355
Peer-to-Peer Activity
Provide pairs of students with a fourth degree polynomial equation and a fifth
degree polynomial equation. Have them work together to determine the number
of possible combinations of types of roots for each equation. Then have them
graph their equations, and use the graphs to help predict which combination of
roots will be the correct combination for each function. Challenge them to solve
the equations to verify their predictions.
355
Lesson 7.2
3/19/14 3:23 PM
Reflect
EXPLAIN 1
1.
Examine the table. For each function, count the number of unique zeros, both real and
non-real. How does the number of unique zeros compare with the degree?
The number of unique zeros is less than or equal to the degree.
2.
Examine the table again. This time, count the total number of zeros for each function,
where a zero of multiplicity m is counted as m zeros. How does the total number of zeros
compare with the degree?
The total number of zeros is the same as the degree of the function.
3.
Discussion Describe the apparent relationship between the degree of a polynomial
function and the number of zeros it has.
The number of zeros of a polynomial function is the same as the degree of the function
Applying the Fundamental Theorem
of Algebra to Solving Polynomial
Equations
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Substantiate The Fundamental Theorem of
when you include complex zeros and count the multiplicities of the zeros in the total.
Explain 1
Applying the Fundamental Theorem of
Algebra to Solving Polynomial Equations
Algebra and its corollary by applying them to
solutions of linear equations and easily factorable
quadratic equations, with which students are familiar.
Include examples of quadratic equations that have
roots with multiplicity of 2.
The Fundamental Theorem of Algebra and its corollary summarize what you have observed earlier
while finding rational zeros of polynomial functions and in completing the Explore.
The Fundamental Theorem of Algebra
Every polynomial function of degree n ≥ 1 has at least one zero, where a zero
may be a complex number.
Corollary: Every polynomial function of degree n ≥ 1 has exactly n zeros,
including multiplicities.
Because the zeros of a polynomial function p(x) give the roots of the equation p(x) = 0, the theorem
and its corollary also extend to finding all roots of a polynomial equation.
Example 1
© Houghton Mifflin Harcourt Publishing Company

Solve the polynomial equation by finding all roots.
2x 3 - 12x 2 - 34x + 204 = 0
The polynomial has degree 3, so the equation has exactly 3 roots.
2x 3 - 12x 2 - 34x + 204 = 0
x 3 - 6x 2 - 17x + 102 = 0
Divide both sides by 2.
(x 3 - 6x 2) - (17x - 102) = 0
Group terms.
Factor out common monomials.
x 2(x - 6) - 17(x - 6) = 0
(x 2 - 17)(x - 6) = 0
Factor out the common binomial.
_
One root is x = 6. Solving x 2 - 17 = 0 gives x 2 = 17, or x = ±√17 .
_
_
The roots are -√17 , √17 , and 6.
Module 7
356
Lesson 2
DIFFERENTIATE INSTRUCTION
A2_MNLESE385894_U3M07L2.indd 356
3/19/14 3:23 PM
Communicating Math
Understanding the concept of the degree of a polynomial is important in applying
the Fundamental Theorem of Algebra and its corollary. Students (especially
English language learners) may benefit from a rigorous review of finding degrees
of polynomials written in standard form, factored form, and with terms in varying
orders of degree. Focus on polynomials that contain only single-variable
monomials. Check that students can explain how to find the degree of the
polynomial for each of the different forms.
Finding Complex Solutions of Polynomial Equations
356
B
QUESTIONING STRATEGIES
x 4 - 6x 2 - 27 = 0
The polynomial has degree 4 , so the equation has exactly 4 roots.
If, after using synthetic substitution to test all
possible rational roots of a cubic equation, you
find only one root of the equation, can you conclude
that the remaining roots are imaginary? Explain. No.
The remaining roots may be imaginary or they may
be irrational.
Notice that x 4 - 6x 2 - 27 has the form u 2 - 6u - 27, where u = x 2. So, you
can factor it like a quadratic trinomial.
x 4 - 6x 2 - 27 = 0
(x - 9 )(x + 3 ) = 0
2
Factor the trinomial.
(x + 3 )(x - 3 )(x + 3) = 0
2
Factor the difference of squares.
The real roots are
-3 and
2
3
――
_
x = ±√-3 = ± i  3 .
3 , i √―
3
-3, 3, -i √―
The roots are
. Solving x 2 + 3 = 0 gives x 2 = -3 , or
.
Reflect
4.
Restate the Fundamental Theorem of Algebra and its corollary in terms of the roots of
equations.
Theorem: For every polynomial of degree n ≥ 1, the equation p(x) = 0 has at
least one root, where a root may be a complex number. Corollary: For every
polynomial of degree n ≥ 1, the equation p(x) = 0 has exactly n roots, when
you include multiplicity.
Your Turn
© Houghton Mifflin Harcourt Publishing Company
Solve the polynomial equation by finding all roots.
5.
8x 3 - 27 = 0
6.
(2x - 3)(4x 2 + 6x + 9) = 0
x(x 3 - 13x 2 + 55x - 91) = 0
2x - 3 = 0
One root is x = 0.
3
x=_
Possible rational roots: ±1, ±7, ±13, ±91.
2
4x 2 + 6x + 9 = 0
-(6) ±
―――――
Use synthetic division to test for roots.
√(6)2 - 4(4)(9)
x = ____________________
A second root is x = 7.
2(4)
-6 ± √――
-108
-6 ± 6i √―
3
x = ___________
= ________
8
8
―
-3 ± 3i √3
3 √―
3
x = ________
, or -_
±_
i 3
4
4
4
―
√
-3
+
3i
3
-3
- 3i √―
3
3
_ ________
________
The roots are 2,
p(x) = x 4 - 13x 3 + 55x 2 - 91x
4
, and
4
Solve x 2 - 6x + 13 = 0.
――
2
-(-6) ± (-6) - 4(1)(13)
x = _______________________
2∙1
――
6 ± 4i
= _____
x = _________
2
2
6 ± √-16
.
x = 3 ± 2i
The roots are 0, 7, 3 + 2i, and 3 - 2i.
Module 7
357
Lesson 2
LANGUAGE SUPPORT
A2_MNLESE385894_U3M07L2 357
Communicate Math
Have students work in pairs. Have them write the theorems in this module for
solving polynomial equations, the Rational Zero Theorem, Rational Roots
Theorem, and the Fundamental Theorem of Algebra, and then work together to
explain the theorems in their own words. Then have students write the
explanations and give an example for each theorem.
357
Lesson 7.2
6/27/14 10:53 PM
Explain 2
Writing a Polynomial Function From Its Zeros
EXPLAIN 2
You may have noticed in finding roots of quadratic and polynomial equations that any irrational or complex roots
come in pairs. These pairs reflect the “±” in the quadratic formula. For example, for any of the following number
pairs, you will never have a polynomial equation for which only one number in the pair is a root.
_
√5
―
―
―
―
Writing a Polynomial Function From
its Zeros
_
1 i √3 and _
1 i √3
11 + _
11 - _
and -√ 5 ; 1 + √7 and 1 - √7 ; i and -i; 2 + 14i and 2 - 14i; _
6
6
6
6
―
―
The irrational root pairs a + b √c and a - b √c are called irrational conjugates. The complex root pairs a + bi and
a - bi are called complex conjugates.
Irrational Root Theorem
QUESTIONING STRATEGIES
―
If a polynomial p(x) has rational coefficients and a + b √c is a root of the equation
p(x) = 0, where a and b are rational and √c is irrational, then
a - b √c is also a root of p(x) = 0.
―
―
If one zero of a fourth degree polynomial
function is rational, what must be true about
the other three zeros? One of the three must also be
rational. The other two could be either irrational
conjugates or imaginary conjugates.
Complex Conjugate Root Theorem
If a + bi is an imaginary root of a polynomial equation with real-number coefficients,
then a - bi is also a root.
Is it possible for a fifth degree polynomial
equation to have no real zeros? Explain. No.
Since imaginary zeros occur in conjugate pairs,
there could be at most 4 imaginary zeros. Therefore,
at least one zero must be real.
Because the roots of the equation p(x) = 0 give the zeros of a polynomial function, corresponding theorems apply
to the zeros of a polynomial function. You can use this fact to write a polynomial function from its zeros. Because
irrational and complex conjugate pairs are a sum and difference of terms, the product of irrational conjugates is
always a rational number and the product of complex conjugates is always a real number.
_
_
_
(2 - √10 )(2 + √10 ) = 22 - (√10 )2 = 4 - 10 = -6
_
_
_
(1 - i√2 )(1 + i√2 ) = 12 - (i√2 )2 = 1 - (-1)(2) = 3
Example 2

Write the polynomial function with least degree and a leading coefficient
of 1 that has the given zeros.
AVOID COMMON ERRORS
_
5 and 3 + 2√7
Multipy the conjugates.
Combine like terms.
Simplify.
_
_
p(x) = ⎡⎣x - (3 + 2√7 )⎤⎦ ⎡⎣x - (3 - 2√7 )⎤⎦ (x - 5)
= ⎡⎣x 2 - (3 - 2√7 )x - ( 3 + 2√7 ) x + ( 3 + 2√7 )( 3 - 2√7 )⎤⎦(x - 5)
_
_
_
_
= ⎡⎣x 2 - (3 - 2√7 )x - (3 + 2√7 )x + (9 - 4 ⋅ 7)⎤⎦(x - 5)
_
_
= ⎡⎣x 2 + (-3 + 2√7 - 3 - 2√7 )x + (-19)⎤⎦(x - 5)
_
_
= ⎡⎣x 2- 6x - 19⎤⎦(x - 5)
Distributive property
= x(x 2 - 6x - 19) - 5(x 2 - 6x - 19)
Multiply.
= x 3 - 6x 2 - 19x - 5x 2 + 30x + 95
Combine like terms.
= x 3 - 11x 2 + 11x + 95
―
© Houghton Mifflin Harcourt Publishing Company
Multiply the first
two factors using FOIL.
Students may make errors when multiplying
factors of the form (x - a), where a is an irrational
number such as 3 + √2 or an imaginary number
such as 1 - 4i. Encourage them to multiply each of
these types of factors with the factor that contains the
conjugate of the irrational or imaginary number first,
and show them how to use grouping to make the
multiplication easier.
_
Because irrational zeros come in conjugate pairs, 3 - 2√7 must also be a zero of
the function. Use the 3 zeros to write a function in factored form, then multiply to
write it in standard form.
The polynomial function is p(x) = x 3 - 11x 2 + 11x + 95.
Module 7
A2_MNLESE385894_U3M07L2 358
358
Lesson 2
6/27/14 10:54 PM
Finding Complex Solutions of Polynomial Equations
358
B
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Have students discuss how they could write
2, 3 and 1- i
Because complex zeros come in conjugate pairs, 1 + i must also be a zero of the function.
Use the 4 zeros to write a function in factored form, then multiply to write it in
standard form.
⎡
(
)⎦
⎤
p(x) = ⎡⎣x - (1 + i)⎤⎦ ⎢x - 1 - i ⎥ (x - 2)(x - 3)
⎣
―
the rule for a third degree polynomial function whose
graph passes through (1 + √2 , 0) and the origin.
Then have them find the function, and use a
graphing calculator to check their work.
Multiply the first
two factors using FOIL.
Multipy the conjugates.
(
)
⎤
⎡
= ⎢x 2 - (1 - i)x - 1 + i x + (1 + i)(1 - i)⎥ (x - 2)(x - 3)
⎣
⎦
( ( ))
⎡
= ⎢x 2 - (1 - i)x - (1 + i)x + 1 - -1
⎣
⎤
⎥(x - 2)(x - 3)
⎦
Combine like terms.
2
= ⎡⎣x + (-1 + i - 1 - i)x + 2⎤⎦(x - 2)(x - 3)
Simplify.
= x 2 - 2x + 2 (x - 2)(x - 3)
Multipy the binomials.
= (x 2 - 2x + 2)
Distributive property
(
)
(x 2 - 5x + 6)
= x2(x2 - 5x + 6) - 2x (x2 - 5x + 6) + 2(x2 - 5x + 6)
Multipy.
= (x4 - 5x3 + 6x2) + (-2x 3 + 10x2 - 12x) + (2x 2 - 10x + 12)
Combine like terms.
= x 4 - 7x 3 + 18x 2 - 22x + 12
4
3
2
The polynomial function is p(x) = x - 7x + 18x - 22x + 12 .
Reflect
7.
―
Restate the Irrational Root Theorem in terms of the zeros of polynomial functions.
If a polynomial function p(x) has rational coefficients and a + b√c is a zero of
―
―
© Houghton Mifflin Harcourt Publishing Company
the function, where a and b are rational and √c is irrational, then a - b√c is
also a zero of p(x).
8.
Restate the Complex Conjugates Zero Theorem in terms of the roots of equations.
If a + bi is an imaginary zero of a polynomial function p(x) with real-number
coefficients, then a - bi is also a zero of p(x).
Module 7
A2_MNLESE385894_U3M07L2 359
359
Lesson 7.2
359
Lesson 2
6/27/14 10:54 PM
Your Turn
EXPLAIN 3
Write the polynomial function with the least degree and a leading
coefficient of 1 that has the given zeros.
9.
_
2 + 3i and 4 - 7√ 2
―
Solving a Real-World Problem by
Graphing Polynomial Functions
The polynomial function must also have 2 - 3i and 4 + 7 √2 as zeros.
p(x) = ⎡⎣x - (2 + 3i)⎤⎦⎡⎣x - (2 - 3i)⎤⎦⎡⎣x - (4 + 7√2 )⎤⎦⎡⎣x - (4 - 7√2 )⎤⎦
= ⎡⎣x 2 - (2 - 3i)x - (2 + 3i)x + (2 + 3i)(2 - 3i)⎤⎦⎡⎣x 2 - (4 - 7√2 )x - (4 + 7√2 )
x + (4 + 7√2 )(4 - 7√2 )⎤⎦
= ⎡⎣x 2 - (2 - 3i)x - (2 + 3i)x + (4 - 9(-1))⎤⎦⎡⎣x 2 - (4 - 7√2 )x - (4 + 7√2 )
x + (16 - 49 ∙ 2)⎤⎦
= ⎡⎣x 2 + (-2 + 3i - 2 - 3i)x + 13⎤⎦⎡⎣x 2 + (-4 + 7√2 - 4 - 7√2 )x - 82⎤⎦
―
―
―
―
―
―
―
―
= (x 2 - 4x + 13)(x 2 - 8x - 82)
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Lead students to recognize that the solution
―
―
of the problem is not a zero of either p(x) or q(x);
however, it is a zero of the difference function p(x)
- q(x). This can be confirmed from the graphs of the
three functions.
= x 2(x 2 - 8x - 82) -4x(x 2 - 8x - 82) + 13(x 2 - 8x - 82)
= (x 4 - 8x 3 - 82x 2) + (-4x 3 + 32x 2 + 328x) + (13x 2 - 104x - 1066)
= x 4 - 12x 3 - 37x 2 + 224x - 1066
The polynomial function is p(x) = x 4 - 12x 3 - 37x 2 + 224x - 1066.
Explain 3
Solving a Real-World Problem by
Graphing Polynomial Functions
You can use graphing to help you locate or approximate any real zeros of a
polynomial function. Though a graph will not help you find non-real zeros, it
can indicate that the function has non-real zeros. For example, look at the graph
of p(x) = x 4 - 2x 2 - 3.
Module 7
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y
2
x
-4
-2
0
-2
2
4
© Houghton Mifflin Harcourt Publishing Company
The graph intersects the x-axis twice, which shows that the function has two
real zeros. By the corollary to the Fundamental Theorem of Algebra, however,
a fourth degree polynomial has_four
_ zeros. So, the other two zeros of p(x) must
be non-real. The zeros are -√3 , √3 , i, and -i. (A polynomial whose graph has a
turning point on the x-axis has a real zero of even multiplicity at that point. If the
graph “bends” at the x-axis, there is a real zero of odd multiplicity greater than 1
at that point.)
4
Lesson 2
3/19/14 3:22 PM
Finding Complex Solutions of Polynomial Equations
360

QUESTIONING STRATEGIES
Why do the methods shown in Parts A and B
produce the same solution? When you solve
the equation p(x) = q(x), you are finding the value
of x for which the two functions are equal. Since p(x)
is equal to q(x) at this value of x, this is the value
that would make their difference, p(x) - q(x), equal
to 0.
The following polynomial models approximate the total oil consumption C (in millions of
barrels per day) for North America (NA) and the Asia Pacific region (AP) over the period
from 2001 to 2011, where t is in years and t = 0 represents 2001.
C NA(t) = 0.00494t 4 - 0.0915t 3 + 0.442t 2 - 0.239t + 23.6
C AP(t) = 0.00877t 3 - 0.139t 2 + 1.23t + 21.1
Use a graphing calculator to plot the functions and approximate the x-coordinate of
the intersection in the region of interest. What does this represent in the context of
this situation? Determine when oil consumption in the Asia Pacific region overtook
oil consumption in North America using the requested method.
Graph Y1 = 0.00494x 4 - 0.0915x 3 + 0.442x 2 - 0.239x + 23.6 and
Y2 = 0.00877x 3 - 0.139x 2 + 1.23x + 21.1. Use the “Calc” menu to find
the point of intersection. Here are the results for Xmin = 0, Xmax = 10,
Ymin = 20, Ymax = 30. (The graph for the Asia Pacific is the one that
rises upward on all segments.)
The functions intersect at about x = 5, which represents the year 2006.
This means that the models show oil consumption in the Asia Pacific
equaling and then overtaking oil consumption in North America about 2006.

Find a single polynomial model for the situation in Example 3A whose zero
represents the time that oil consumption for the Asia Pacific region overtakes
consumption for North America. Plot the function on a graphing calculator and use
it to find the x-intercept.
Let the function C D(t) represent the difference in oil consumption in the Asia Pacific
and North America.
A difference of 0 indicates the time that consumption is equal .
C D(t)
=
C AP(t)
- C NA(t)
= 0.00877t - 0.139t 2 + 1.23t + 21.1 - (0.00494t 4 - 0.0915t 3 + 0.442t 2 - 0.239t + 23.6)
3
© Houghton Mifflin Harcourt Publishing Company
Remove parentheses and rearrange terms.
= -0.00494t 4 + 0.00877t 3 + 0.0915t 3 - 0.139t 2 - 0.442t 2 + 1.23t + 0.239t + 21.1 - 23.6
Combine like terms. Round to three significant digits.
= -0.00494t 4 + 0.100t 3 - 0.581t 2 + 1.47t - 2.50
Graph C D(t) and find the x-intercept. (The graph with Ymin = -4, Ymax = 6 is shown.)
Within the rounding error, the results for the x-coordinate
of the intersection of C NA(t) and C AP(t) and the x-intercept
of C D(t) are the same.
Module 7
A2_MNLESE385894_U3M07L2 361
361
Lesson 7.2
361
Lesson 2
16/10/14 11:07 AM
Your Turn
ELABORATE
10. An engineering class is designing model rockets for a competition. The body of the rocket
must be cylindrical with a cone-shaped top. The cylinder part must be 60 cm tall, and the
height of the cone must be twice the radius. The volume of the payload region must
be 558π cm 3 in order to hold the cargo. Use a graphing calculator to graph the rocket’s
payload volume as a function of the radius x. On the same screen, graph the constant
function for the desired payload. Find the intersection to find x.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Lead students to the generalization that a
Let V represent the volume of the payload region.
V = Vcone + Vcylinder
V(x) =
_1 πx (2x) + πx (60) = _2 πx
2
2
3
3
3
To find x when the volume is 558π, graph
2
y = πx 3 + 60πx 2 and y = 558π and
3
find the points of intersection.
polynomial function of odd degree must have an odd
number (counting repeated zeros) of real zeros and,
in particular, must have at least one real zero.
+ 60πx 2
_
QUESTIONING STRATEGIES
Because the radius must be positive, the radius of the rocket is 3 cm.
A fourth degree polynomial function has only
the zeros -2, 3, and 4. How can this be true
given the requirement of the Corollary of the
Fundamental Theorem of Algebra, which states that a
polynomial of degree n has exactly n zeros? One of
the zeros must occur twice. The corollary requires
that repeated zeros be counted multiple times.
Elaborate
11. What does the degree of a polynomial function p(x) tell you about the zeros of the function
or the roots of the equation p(x) = 0?
The degree tells you how many zeros or roots there are when you include complex zeros or
roots and count the multiplicities of repeated zeros or roots.
12. A polynomial equation of degree 5 has the roots 0.3, 2, 8, and 10.6 (each of multiplicity 1).
What can you conclude about the remaining root? Explain your reasoning.
The remaining root must be rational. This is because any irrational roots or imaginary
roots always occur in conjugate pairs. So, if there were an irrational or imaginary root,
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Ask students to discuss the possibility of two
there would have to be two of them.
point where the two graphs intersect. Also, you can form a new function that is the
difference of the two original functions. The x-intercepts of the graph of this function will
also be the x-values where the original functions have the same value.
14. Essential Question Check-In What are possible ways to find all the roots of a polynomial equation?
By the corollary to the Fundamental Theorem of Algebra, you know that the number
© Houghton Mifflin Harcourt Publishing Company
13. Discussion Describe two ways you can use graphing to determine when two polynomial
functions that model a real-world situation have the same value.
You can graph both functions on the same coordinate grid and find the x-value of any
polynomial functions that model a real-world
situation having more than one value for which they
are equal. Have them discuss the implications of this
situation on the graphs of the functions and on the
graph of the difference function.
of roots equals the degree of the equation. You can factor when possible, and use the
Rational Root theorem along with the Zero Product Property to find rational roots. You can
SUMMARIZE THE LESSON
use the quadratic formula to find irrational or complex roots.
Module 7
A2_MNLESE385894_U3M07L2.indd 362
362
Lesson 2
6/28/14 12:37 AM
How can you use the Fundamental Theorem
of Algebra, its corollary, and the Irrational
Conjugates and Complex Conjugates Theorems to
determine the possible combinations of types of zeros
of a polynomial function? You can use the
Fundamental Theorem of Algebra and its corollary
to find the total number of zeros of the function.
Then you can use the fact that irrational and
imaginary zeros occur in conjugate pairs to
determine the possible combinations.
Finding Complex Solutions of Polynomial Equations
362
EVALUATE
Evaluate: Homework and Practice
• Online Homework
• Hints and Help
• Extra Practice
Find all zeros of p(x). Include any multiplicities greater than 1.
1.
p(x) = 3x 3 - 10x 2 + 10x - 4
2.
1
Possible rational zeros are ±1, ±2, ±4, ±_
,
3
2
4
_
_
± ,± .
3
ASSIGNMENT GUIDE
3
Concepts and Skills
Practice
Explore
Investigating the Number of
Complex Zeros of a Polynomial
Function
Exercises 1–2
Example 1
Applying the Fundamental Theorem
of Algebra to Solving Polynomial
Equations
Exercises 3–4
Example 2
Writing a Polynomial Function
From its Zeros
Exercises 5–8
Example 3
Solving a Real-World Problem by
Graphing Polynomial Functions
Exercises 9–11
= x 2(x - 3) + 4(x - 3)
= (x 2 + 4)(x - 3)
x=
――――――
3 is a zero.
(-4) ± √(-4) - 4(3)(2)
___________________
2
――
2(3)
―
―
= _______
= ______
x = _______
6
6
3
4 ± √-8
4 ± 2i √2
2 ± i √2
2
+ i √―
2
______
3
, and
Solve x 2 + 4 = 0.
x 2 = -4
The zeros of p(x) are 3, -2i, and 2i.
.
3
――
x = ±√-4 = ±2i
2
- i √―
2
______
Solve the polynomial equation by finding all roots.
3.
2x 3 - 3x 2 + 8x - 12 = 0
4.
x 2 (2x - 3) + 4(2x - 3) = 0
2
0 is a root.
+ 4)(2x - 3) = 0
Possible rational roots are 1 and -1.
2x - 3 = 0
3
x=_
1 is a root.
x(x - 1)(x 2 - 4x - 1) = 0
2
x2 + 4 = 0
© Houghton Mifflin Harcourt Publishing Company
x 2 = -4
Solve x 2 - 4x - 1 = 0.
-(-4) ±
――
Exercise
2
2
―
―
The roots are 0, 1, 2 + √5 , and 2 - √5 .
Lesson 2
363
Depth of Knowledge (D.O.K.)
2(1)
4 ± √―
20
4 ± 2 √―
5
5
x = ______ = ______ = 2 ± √―
3
The roots are _
, -2i, and 2i.
2
Module 7
――――――
√(-4)2 - 4(1)(-1)
x = ____________________
x = ±√-4 = ±2i
A2_MNLESE385894_U3M07L2.indd 363
x 4 - 5x 3 + 3x 2 + x = 0
x(x 3 - 5x 2 + 3x + 1) = 0
(2x 3 - 3x 2) + (8x - 12) = 0
(x
How does the Rational Zero Theorem help
you find zeros that are not rational? The
Rational Zero Theorem can be used to identify
rational zeros and the corresponding factors. Then,
other methods, such as the quadratic formula, may
be used to find other zeros that are irrational or
imaginary.
Lesson 7.2
= (x 3 - 3x 2) + (4x - 12)
Solve 3x 2 - 4x + 2 = 0.
The zeros of p(x) are 2,
QUESTIONING STRATEGIES
363
p(x) = x 3 - 3x 2 + 4x - 12
p(x) = (x - 2)(3x 2 - 4x + 2)
2 is a zero.
p(x) = x 3 - 3x 2 + 4x - 12
Mathematical Practices
1–8
2 Skills/Concepts
MP.2 Reasoning
9–10
2 Skills/Concepts
MP.6 Precision
11
2 Skills/Concepts
MP.4 Modeling
12
3 Strategic Thinking
MP.2 Reasoning
13–14
3 Strategic Thinking
MP.2 Reasoning
15
3 Strategic Thinking
MP.2 Reasoning
3/19/14 3:22 PM
Write the polynomial function with least degree and a leading
coefficient of 1 that has the given zeros.
5.
0,
_
√5 ,
and 2
6.
―
Because irrational zeros come in conjugate
pairs, -√5 must also be a zero.
―
―
p(x) = x(x - √5 )(x + √5 )(x - 2)
= x(x - 5)(x - 2)
2
7.
3
Because complex zeros come in conjugate
pairs, -4i must also be a zero.
p(x) = (x - 2)(x + 2)(x - 4i)(x + 4i)
= x 4 + 12x 2 - 64
2
= x - 2x - 5x + 10x
4
―
Students often make sign errors when writing factors
for zeros or roots that are irrational, such as 2 - √5 ,
or imaginary, such as 2 + i. Encourage them to use
parentheses within parentheses when writing the
factors, and to be careful to apply the distributive
property when removing the parentheses or
regrouping the terms.
4i, 2, and -2
= (x 2 - 4)(x 2 + 16)
= x(x - 2x - 5x + 10)
3
AVOID COMMON ERRORS
2
1, -1 (multiplicity 3), and 3i
Because complex zeros come in conjugate pairs, -3i must also be a zero.
3
p(x) = (x - 1)(x + 1) (x - 3i)(x + 3i)
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 Have students discuss why irrational roots of
2
= ⎡⎣(x - 1)(x + 1)⎤⎦ (x + 1) (x - 3i)(x + 3i)
= (x 2 - 1)(x 2 + 2x + 1)(x 2 + 9)
= (x 4 + 8x 2 - 9)(x 2 + 2x + 1)
= x 4 (x 2 + 2x + 1) + 8x 2 (x 2 + 2x + 1) - 9(x 2 + 2x +1)
a polynomial equation with rational coefficients must
occur in conjugate pairs. Have them consider the
resulting polynomial if, for example, only one of three
factors of a cubic polynomial equation contained an
irrational number.
= x 6 + 2x 5 + x 4 + 8x 4 + 16x 3 + 8x 2 - 9x 2 - 18x - 9
= x 6 + 2x 5 + 9x 4 + 16x 3 - x 2 - 18x - 9
8.
3(multiplicity of 2) and 3i
Because complex zeros come in conjugate pairs, -3i must also be a zero.
2
p(x) = (x - 3) (x - 3i)(x + 3i)
= (x 2 - 6x + 9)(x 2 + 9)
© Houghton Mifflin Harcourt Publishing Company
= x 2(x 2 - 6x + 9) + 9(x 2 - 6x + 9)
= x 4 - 6x 3 + 9x 2 + 9x 2 - 54x + 81
= x 4 - 6x 3 + 18x 2 - 54x + 81
Module 7
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Lesson 2
3/19/14 3:22 PM
Finding Complex Solutions of Polynomial Equations
364
9.
VISUAL CUES
Have students graph several of the functions using a
graphing calculator to provide a visual connection
between each type of zero (rational, irrational, and
imaginary), and its representation on the graph of the
function. Help students to see how irrational zeros
can be approximated from x-intercepts. Lead them to
observe that a function that has only imaginary zeros
has no x-intercepts.
Forestry Height and trunk volume measurements from
10 giant sequoias between the heights of 220 and 275 feet
in California give the following model, where h is the
height in feet and V is the volume in cubic feet.
V(h) = 0.131h 3 - 90.9h 2 + 21,200h - 1,627,400
The “President” tree in the Giant Forest Grove in Sequoia
National Park has a volume of about 45,100 cubic feet. Use
a graphing calculator to plot the function V(h) and the
constant function representing the volume of the President
tree together. (Use a window of 220 to 275 for X and
30,000 to 55,000 for Y.) Find the x-coordinate of the intersection of the graphs. What
does this represent in the context of this situation?
The x-coordinate of the intersection gives
the model’s predicted height for a tree
with the volume of the President tree. This
predicted height is about 265 feet.
CRITICAL THINKING
Students may be interested to find that they can test
irrational and imaginary zeros of a polynomial
function using synthetic substitution. Encourage
them to use this process to check their work.
10. Business Two competing stores, store A and store B, opened the same year in the
same neighborhood. The annual revenue R (in millions of dollars) for each store t
years after opening can be approximated by the polynomial models shown.
R A(t) = 0.0001(-t 4 + 12t 3 - 77t 2 + 600t + 13,650)
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©RichardBakerUSA/Alamy
R B(t) = 0.0001(-t 4 + 36t 3 - 509t 2 + 3684t + 3390)
Using a graphing calculator, graph the models from t = 0 to t = 10,
with a range of 0 to 2 for R. Find the x-coordinate of the intersection
of the graphs, and interpret the graphs.
Graph Y1 = 0.0001(-x 4 + 12x 3 - 77x 2 + 600x + 13,650) for R A.
Graph Y2 = 0.0001(-x 4 + 36x 3 - 509x 2 + 3684x + 3390) for R B.
Then find the point of intersection.
The functions intersect at x = 9, which corresponds to having
the same annual revenue 9 years after the stores opened.
Module 7
A2_MNLESE385894_U3M07L2 365
365
Lesson 7.2
365
Lesson 2
6/27/14 11:03 PM
11. Personal Finance A retirement account contains cash and stock in a company.
The cash amount is added to each week by the same amount until week 32, then that
same amount is withdrawn each week. The functions shown model the balance B
(in thousands of dollars) over the course of the past year, with the time t in weeks.
LANGUAGE SUPPORT
Connect Vocabulary
Remind students that they learned complex numbers
have a real and an imaginary part. The complex
conjugate of a + bi is a - bi, and similarly the
complex conjugate of a - bi is a + bi. This consists
of changing the sign of the imaginary part of a
complex number. The real part is left unchanged.
B C(t) = -0.12|t - 32| + 13
B S(t) = 0.00005t 4 - 0.00485t 3 + 0.1395t 2 - 1.135t + 15.75
Use a graphing calculator to graph both models (Use 0 to 20 for range.). Find the
x-coordinate of any points of intersection. Then interpret your results in the context of this
situation.
The graphs intersect at x-values of about 38 and 47.
This means that at those weeks of the year, the cash
balance and stock balance in the account were the
same.
12. Match the roots with their equation.
A, B, E, F
A. 1
B. -2
x 4 + x 3 + 2x 2 + 4x - 8 = 0
A, B, C, D x 4 - 5x 2 + 4 = 0
C. 2
D. -1
E. 2i
F. -2i
x 4 + x 3 + 2x 2 + 4x - 8 = 0 in factored form is (x - 1)(x + 2)(x 2 + 4) = 0.
Roots are 1, -2, 2i, and -2i.
Module 7
A2_MNLESE385894_U3M07L2.indd 366
366
© Houghton Mifflin Harcourt Publishing Company
x 4 - 5x 2 + 4 = 0 in factored form is (x + 1)(x - 1)(x + 2)(x - 2) = 0.
Roots are -1, 1, -2, and 2.
Lesson 2
3/19/14 3:21 PM
Finding Complex Solutions of Polynomial Equations
366
PEER-TO-PEER DISCUSSION
H.O.T. Focus on Higher Order Thinking
13. Draw Conclusions Find all of the roots of x 6− 5x − 125x 2 + 15,625 = 0. (Hint:
Rearrange the terms with a sum of cubes followed by the two other terms.)
4
Ask students to discuss with a partner why, although
the Rational Root Theorem can always be used to
help find the roots of a cubic equation, it may not be
useful for finding the roots of a fourth degree
polynomial equation. Since a cubic equation has
three roots, at least one of them will be rational
(since irrational and imaginary roots occur in
conjugate pairs). The other two roots, no matter
what type, can be found by factoring or by using the
quadratic formula. A fourth degree equation will
have four roots, none of which may be rational, so
the Rational Root Theorem will not be of help.
(x 6 + 15,625)- 25x 4 - 625x 2 = 0
⎡(x 2) 3+ 25 3⎤ - 25x 4 - 625x 2 = 0
⎣
⎦
(x 2 + 25)(x 4 - 25x 2 + 625) - 25x 2 (x 2 + 25)= 0
(x 2 + 25)(x 4 - 25x 2 + 625 - 25x 2) = 0
(x 2 + 25)(x 4 -50x 2 + 625) = 0
(x 2 + 25)(x 2 - 25) 2 = 0
2
(x 2 + 25)⎡⎣(x+ 5)(x - 5)⎤⎦ = 0
The roots are -5 and 5, each with multiplicity 2, and -5i and 5i.
14. Explain the Error A student is asked to write the polynomial function
_ with least
degree and a leading coefficient of 1 that has the zeros 1 + i, 1 - i, √2 , and -3. The
student writes the product
of factors
them
together to obtain
_
_ shown, and multiplies
_
_
p(x) = x 4 + (1 - √2 )x 3 - (4 + √2 )x 2 + (6 + 4√2 )x - 6√2 . What error did the
student make? What is the correct function?
―
―
The function must have 5 zeros. The zero √2 must be paired with its conjugate, - √2 .
p(x) = ⎡⎣x - (1 + i)⎤⎦⎡⎣x - (1 - i)⎤⎦(x - √2 )(x + √2 )(x + 3)
= ⎡⎣x 2 - (1 - i)x -(1 + i)x + (1 + i)(1 - i)⎤⎦(x 2 - 2)(x + 3)
= ⎡⎣x 2 + (-1 + i -1 - i) x + (1 -(-1))⎤⎦(x 3 + 3x 2 -2x - 6)
= (x 2 - 2x + 2)(x 3 + 3x 2 - 2x - 6)
JOURNAL
―
Have students describe how they would go about
finding the roots of a fifth degree polynomial
equation if they know that at least two of the roots
are rational.
―
= (x 5 + 3x 4 - 2x 3 - 6x 2) + (-2x 4 - 6x 3 + 4x 2 + 12x) + (2x 3 + 6x 2 - 4x - 12)
© Houghton Mifflin Harcourt Publishing Company
= x 5 + x 4 - 6x 3 + 4x 2 + 8x - 12
of a polynomial equation that has 3i as a
15. Critical Thinking What is the least degree
_
root with a multiplicity of 3, and 2 - √3 as a root with multiplicity 2? Explain.
―
Module 7
A2_MNLESE385894_U3M07L2.indd 367
367
Lesson 7.2
―
The least degree is 10. Since 3i is a root 3 times, then -3i must also be a
root 3 times. Since 2 - √3 is a root 2 times, then 2 + √3 must also be a
root 2 times, and 3 + 3 + 2 + 2 = 10.
367
Lesson 2
3/19/14 3:21 PM
Lesson Performance Task
CONNECT VOCABULARY
Students may not be familiar with the abbreviations
of the movie rating system. Explain that the
abbreviations indicate how appropriate the movie is
for difference audiences. A G rating means the movie
is for General audiences. A PG rating means Parental
Guidance is suggested. A PG-13 rating means
Parental Guidance is suggested and the movie may
not be appropriate for children under age 13. An R
rating means entrance is Restricted; an adult must
accompany children under 17.
In 1984 the MPAA introduced the PG-13 rating to their movie rating system. Recently,
scientists measured the incidences of a specific type of violence depicted in movies. The
researchers used specially trained coders to identify the specific type of violence in one half
of the top grossing movies for each year since 1985. The trend in the average rate per hour
of 5-minute segments of this type of violence in movies rated G/PG, PG-13, and R can be
modeled as a function of time by the following equations:
V G/PG(t) = -0.015t + 1.45
V PG-13(t) = 0.000577t 3 - 0.0225t 2 + 0.26t + 0.8
V R(t) = 2.15
V is the average rate per hour of 5-minute segments containing the specific
type of violence in movies, and t is the number of years since 1985.
b. What do the equations indicate about the relationship between
V G/PG(t) and V PG-13(t) as t increases?
c. Graph the models for V G/PG(t) and V PG-13(t) and find the year
in which V PG-13(t) will be greater than V G/PG(t).
Rate per Hour
a. Interestingly, in 1985 or t = 0, V G/PG(0) > V PG-13 (0). Can you
think of any reasons why this would be true?
3
2
V(t)
VR
VPG–13
AVOID COMMON ERRORS
VG/PG
1
Students may think that the models V(t) give the
total amount of violence in a movie. Ask students
what the units of V(t) are. number of 5-minute
segments per hour Ask students how to calculate
the total minutes of violence in a movie. Multiply
V(t) by 5 and then multiply by the length of the
movie in minutes.
t
0
6
12 18 24
Years Since 1985
a. Possible answers include but are not limited to
•The rating of PG-13 was poorly understood by the people responsible for
rating the films.
© Houghton Mifflin Harcourt Publishing Company
•Films released in the years immediately following 1985 had been
scripted, filmed, and/or edited before the rating was fully understood by
the film studios, so they hadn’t separated the specific type of violence
out of the G/PG movies.
b. The equations indicate that as t increases, V PG-13(t) will eventually be
greater than V G/PG(t). V G/PG(t) is a linear function with a negative first term
so its end behavior on the right is decreasing to negative infinity while
the leading term of V PG-13(t) is positive, so its end behavior on the right is
increasing to infinity.
c. The functions intersect at a value of t ≈ 3, which indicates that the average
rate per hour of 5-minute segments of violence in movies rated PG-13 first
surpassed the average hourly rate in movies rated G/PG in 1988.
Module 7
368
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 Discuss with students why V PG-13 increases to
infinity as t increases. Ask them if it makes sense that
V PG-13 becomes greater than V R and whether they
think this will actually happen. Have students explain
how they could create a model that would more
accurately predict V PG-13 for future years..
Lesson 2
EXTENSION ACTIVITY
A2_MNLESE385894_U3M07L2.indd 368
Have students research the top-grossing movie for each year since 1985 and
whether it was rated G, PG, PG-13, or R. Have students discuss whether the
success of a movie is related to its rating. Ask them if they think the amount of
violence in a movie makes it more or less popular.
6/28/14 12:44 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 Complex Solutions of Polynomial Equations
368