LESSON
7.1
Name
Adding and
Subtracting
Polynomials
Class
7.1
Adding and Subtracting
Polynomials
Essential Question: How do you add or subtract two polynomials, and what type of
expression is the result?
Explore
The student is expected to:
A2.7.B
Mathematical Processes
A2.1.F
Analyze mathematical relationships to connect and communicate
mathematical ideas.

Language Objective
x 3, a 2 bc 12, 76
2
3
Not monomials: y + 3y - 5y + 10
Students work in pairs to create a “parts of a polynomial” chart.

ENGAGE
View the Engage section online. Discuss the photo
and how the records of maximum and minimum
temperatures provide data for two functions. Have
the students identify the independent and dependent
variables of the two functions. Then preview the
Lesson Performance Task.
Identify the monomials: x 3, y + 3y 2 - 5y 3 + 10, a 2 bc 12, 76
Monomials:
1.B, 2.C.1, 2.C.2, 4.F

© Houghton Mifflin Harcourt Publishing Company
PREVIEW: LESSON
PERFORMANCE TASK
Identifying and Analyzing Monomials
and Polynomials
A polynomial function of degree n has the standard form p(x) = a nx n + a n-1 x n-1 + … + a 2x 2 +
a 1x + a 0, where a n, a n-1,…, a 2, a 1, and a 0 are real numbers. The expression a nx n + a n-1 x n-1 + …
a 2x 2 + a 1x + a 0 is called a polynomial, and each term of a polynomial is called a monomial.
A monomial is the product of a number and one or more variables with whole-number exponents.
A polynomial is a monomial or a sum of monomials. The degree of a monomial is the sum of the
exponents of the variables, and the degree of a polynomial is the degree of the monomial term with
the greatest degree. The leading coefficient of a polynomial is the coefficient of the term with the
greatest degree.
Add, subtract, and multiply polynomials.
First, add or subtract like terms. The sum or
difference is another polynomial.


Identify the degree of
each monomial.
Monomial
x3
a 2 bc 12
76
Degree
3
15
0
2
3
Identify the terms of the polynomial y + 3y 2 - 5y 3 + 10. y , 3y , - 5y , 10
Identify the coefficient
of each term.
Identify the degree of
each term.
-5y 3
Term
y
3y 2
Coefficient
1
3
-5
10
3y 2
-5y 3
10
2
3
0
Term
y
Degree
1

3
Write the polynomial in standard form. -5y + 3y + y + 10

What is the leading coefficient of the polynomial?
Module 7
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Lesson 1
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A2_MTXESE353930_U3M07L1.indd 353
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Lesson 1
353
Module 7
L1.indd
0_U3M07
SE35393
A2_MTXE
Lesson 7.1
10
2

353
Resource
Locker
A2.7.B Add, subtract…polynomials.
Texas Math Standards
Essential Question: How do you add
or subtract two polynomials, and what
type of expression is the result?
Date
353
1/12/15
9:30 PM
1/12/15 9:30 PM
Reflect
EXPLORE
Discussion How can you find the degree of a polynomial with multiple variables in each term?
Find the degree of each term by adding the exponents of each variable. The degree of the
1.
Identifying and Analyzing Monomials
and Polynomials
polynomial is the degree of the term with the highest degree.
Adding Polynomials
Explain 1
To add polynomials, combine like terms.

(4x
2
INTEGRATE TECHNOLOGY
Add the polynomials.
Example 1
- x + 2 + 5x ) + (-x + 6x + 3x
3
4
5x
-x
4
+4x
3
2
Students have the option of completing the
polynomial activity either in the book or
)
4
+2
2

online.
Write in standard form.
+3x 4
+6x 2 -x
___
8x 4 -x 3
+10x 2 -x +2
Align like terms.
Add.
QUESTIONING STRATEGIES
(10x - 18x 3 + 6x 4 - 2) + (-7x 4 + 5 + x + 2x 3)
(6x 4 - 18x 3 + 10x - 2) + (-7x 4 + 2x 3 + x + 5)
Write in standard form.
= 6x 4 - 7x 4
Group like terms.
(
) + ( -18x
3
) ( 10x + x) + (-2 + 5 )
+ 2x 3 +
= -x 4 - 16x 3 + 11x + 3
How do you find the degree of a term
containing one variable with no exponent on
the variable? Why? The degree is 1; a variable such
as x is equivalent to x 1.
Add.
How do you recognize the leading
coefficient? Write the polynomial in standard
form. It is the coefficient of the term with the
highest degree.
Reflect
2.
Is the sum of two polynomials always a polynomial? Explain.
Yes. Because the two addends are the sums of monomials, adding them also results in a
sum of monomials, which is by definition a polynomial.
© Houghton Mifflin Harcourt Publishing Company
Your Turn
Add the polynomials.
3.
(17x 4 + 8x 2 - 9x 7 + 4 - 2x 3 ) + (11x 3 - 8x 2 + 12)
(-9x 7 + 17x 4 - 2x 3 + 8x 2 + 4) + (11x 3 - 8x 2 + 12)
= (-9x 7) +(17x 4) + (-2x 3 + 11x 3) + (-8x 2 - 8x 2) + (4 + 12)
= -9x 7 + 17x 4 + 9x 3 + 16
4.
(-8x + 3x 11 + x 6) + (4x 4 - x + 17)
(3x 11 + x 6 - 8x) + (4x 4 - x + 17)
EXPLAIN 1
Adding Polynomials
AVOID COMMON ERRORS
= (3x ) + (x ) + (4x ) + (-8x -x) + (17)
11
6
4
= 3x 11 + x 6 + 4x 4 - 9x + 17
Module 7
354
Lesson 1
PROFESSIONAL DEVELOPMENT
A2_MTXESE353930_U3M07L1 354
Learning Progressions
21/02/14 7:05 AM
Students often add polynomials using the same
method each time, either horizontally or vertically.
Point out that if the polynomials have many terms,
adding them vertically may prevent errors because
students can line up like terms and leave gaps if terms
of some degrees are missing. If the polynomials have
only a few terms, adding them horizontally may be
more convenient, especially when using mental math.
In this lesson, students extend their earlier work with quadratic and cubic polynomial
functions to explore the arithmetic of polynomials of degree n. A polynomial is an
expression involving a sum of whole-number powers of one or more variables that are
multiplied by coefficients. It has the form
p(x) = a nx n + a n - 1x n - 1 + … + a 2x 2 + a 1x + a 0 where a n, a n - 1, …, a 2, a 1, and a 0 are
real numbers, and each term of the expression is called a monomial. Unless otherwise
specified, the coefficients a i are generally restricted to real numbers, although many of
the key results of this module also hold for polynomials with complex coefficients,
which students will learn in future courses.
Adding and Subtracting Polynomials
354
Explain 2
QUESTIONING STRATEGIES
Subtracting Polynomials
To subtract polynomials, combine like terms.
Is it possible to add three or more
polynomials? Explain how. Yes, you can add
three or more polynomials, either by writing the
polynomials vertically, aligning like terms, and
adding their coefficients, or by writing the
polynomials horizontally, grouping like terms, and
adding their coefficients.
Example 2

Subtract the polynomials.
(12x 3 + 5x - 8x 2 + 19) - (6x 2 - 9x + 3 - 18x 3)
Write in standard form.
Align like terms and add the opposite.
Add.

3
-8x 2
+5x
+19
(-4x 2 + 8x 3 + 19 - 5x 5) - (9 + 2x 2 + 10x 5)
Write in standard form and add the
opposite.
Are the commutative and associative
properties of addition true for the addition of
polynomials? Explain. Yes, the sum is the same
regardless of the order in which polynomials are
added. If three or more polynomials are added, the
sum is the same regardless of how the polynomials
are grouped.
12x
+18x 3 -6x 2
+9x
-3
___
30x 3 -14x 2 +14x
+16
(-5x 5 + 8x 3 - 4x 2 + 19) + (-10x 5 - 2x 2 - 9)
(
Group like terms
5
= -5x 5 - 10x
Add
5
= -15x
) + ( 8x ) + ( -4x
3
2
) (
)
- 2x 2 + 19 - 9
2
+ 8x 3 - 6x + 10
Reflect
5.
Is the difference of two polynomials always a polynomial? Explain.
Yes. Finding the difference of two polynomials is the same as finding the sum of the
first polynomial and the opposite of the second. The sum of two polynomials is always a
polynomial, so the difference of two polynomials is also always a polynomial.
EXPLAIN 2
Your Turn
Subtract the polynomials.
Subtracting Polynomials
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
Point out that subtracting like terms in polynomials
is just like subtracting numbers. For a term with the
same variables and exponents like x 2y 5, which we’ll
call something, it is: 8 something minus 5 something
is 3 something, 6 something minus -4 something is
10 something, and so on. Have students think up
more of these analogous patterns.
QUESTIONING STRATEGIES
Is it necessary to write polynomials in
standard form before subtracting them?
Explain. No, as long as you group like terms before
subtracting and keep track of signs correctly, the
results will be the same. However, it is good practice
to write the difference as a polynomial in standard
form.
355
Lesson 7.1
© Houghton Mifflin Harcourt Publishing Company
6.
(23x 7 - 9x 4 + 1) - (-9x 4 + 6x 2 - 31)
= (23x 7 - 9x 4 + 1) + (9x 4 - 6x 2 - 31)
= (23x 7) + (-9x 4 + 9x 4) + (-6x 2) + (1 - 31)
= 23x 7 - 6x 2 - 30
7.
(7x 3 + 13x - 8x 5 + 20x 2) - (-2x 5 + 9x 2)
(-8x5 + 7x 3 + 20x 2 + 13x) + (2x 5 - 9x 2)
= (-8x 5 + 2x 5) + (7x 3) + (20x 2 - 9x 2) + (13x)
= -6x 5 + 7x 3 + 11x 2 + 13x
Module 7
355
Lesson 1
COLLABORATIVE LEARNING
A2_MTXESE353930_U3M07L1 355
Whole Class Activity
Have groups of students create posters to describe how to add or subtract
polynomials. Have them write an example addition (or subtraction) problem in
the top cell of a graphic organizer. Then have them write each of the steps below it
in one box, with an explanation in words alongside it in another box. Have
students use arrows to connect the steps.
22/02/14 7:59 AM
Explain 3
Modeling with Polynomial Addition and
Subtraction
AVOID COMMON ERRORS
Regardless of the method they use to subtract
polynomials, students may not remember to
distribute the subtraction operation to all terms in
the second polynomial. The result is that the first
term is subtracted while the others are added.
Remind students to always watch for this potential
mistake. Encourage students to check their answers;
just as a numerical difference can be checked by
addition, students can check a polynomial difference
by addition.
Polynomial functions can be used to model real-world quantities. If two polynomial functions model
quantities that are two parts of a whole, the functions can be added to find a function that models the
quantity as a whole. If the polynomial function for the whole and a polynomial function for a part are given,
subtraction can be used to find the polynomial function that models the other part of the whole.
Example 3

Find the polynomial that models the problem and use it to estimate the quantity.
The data from the U.S. Census Bureau for 2005–2009 shows that the number of male students
enrolled in high school in the United States can be modeled by the function M(x) = -10.4x 3
+ 74.2x 2 - 3.4x + 8320.2, where x is the number of years after 2005 and M(x) is the number
of male students in thousands. The number of female students enrolled in high school in the
United States can be modeled by the function F(x) = -13.8x 3 + 55.3x 2 + 141x + 7880, where
x is the number of years after 2005 and F(x) is the number of female students in thousands.
Estimate the total number of students enrolled in high school in the United States in 2009.
In the equation T(x) = M(x) + F(x), T(x) is the total number of students in
thousands.
Add the polynomials.
EXPLAIN 3
(-10.4x 3 + 74.2x 2 - 3.4x + 8320.2) + (-13.8x 3 + 55.3x 2 + 141x + 7880)
= (-10.4x 3 - 13.8x ) + (74.2x 2 + 55.3x 2) + (-3.4x + 141x) + (8320.2 + 7880)
3
Modeling with Polynomial Addition
and Subtraction
= -24.2x 3 + 129.5x 2 + 137.6x + 16,200.2
The year 2009 is 4 years after 2005, so substitute 4 for x.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jutta
Klee/Corbis
3
2
-24.2(4) + 129.5(4) + 137.6(4) + 16,200.2 ≈ 17,274
About 17,274 thousand students were enrolled in high school in the United States
in 2009.

The data from the U.S. Census Bureau for 2000–2010
shows that the total number of overseas travelers visiting
New York and Florida can be modeled by the function
T(x) = 41.5x 3 - 689.1x 2 + 4323.3x + 2796.6, where x is the
number of years after 2000 and T(x) is the total number of
travelers in thousands. The number of overseas travelers
visiting New York can be modeled by the function
N(x) = -41.6x 3 + 560.9x 2 - 1632.7x + 6837.4, where x is the
number of years after 2000 and N(x) is the number of travelers
in thousands. Estimate the total number of overseas travelers to
Florida in 2008.
In the equation F(x) = T(x) - N(x), F(x) is the number of travelers to Florida in thousands.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Modeling
Many real-world situations in fields ranging from
education and business to engineering and physics
can be modeled by polynomial functions over a
restricted domain. Students are provided polynomials
that model real-world situations. Polynomial
functions can also be fit to data for analysis. By
adding and subtracting polynomials, students extend
the model to determine relationships in the data.
Subtract the polynomials.
(41.5x 3 - 689.1x 2 + 4323.3x + 2796.6) - (-41.6x 3 + 560.9x 2 - 1632.7x + 6837.4)
QUESTIONING STRATEGIES
= (41.5x 3 - 689.1x 2 + 4323.3x + 2796.6) + (41.6x 3 - 560.9x 2 + 1632.7x - 6837.4)
Module 7
356
Lesson 1
DIFFERENTIATE INSTRUCTION
Visual Cues
A2_MTXESE353930_U3M07L1 356
Have students use color coding to circle and identify the like terms when adding
or subtracting polynomials. They can also include arrows to help them identify
like terms when adding polynomials in horizontal form.
15-01-11 2:55 AM
How can adding or subtracting polynomials
help you model a real-world
quantity? Sample answer: Since the sum or
difference of two or more polynomials is a
polynomial, then the sums or differences of
polynomials that model quantities can also can
model a quantity.
Multiple Representations
Have students make a poster showing the methods for adding and subtracting
polynomials horizontally and vertically. For each method, students should provide
an example.
Adding and Subtracting Polynomials
356
(
= 41.5x 3 + 41.6x 3
ELABORATE
= 83.1 x 3 -
) + ( -689.1x
1250 x 2 +
2
) (
) (
- 560.9x 2 + 4323.3x + 1632.7x + 2796.6 - 6837.4
5956 x - 4040.8
)
The year 2008 is 8 years after 2000, so substitute 8 for x.
QUESTIONING STRATEGIES
3
2
83.1(8) - 1250(8) + 5956(8) - 4040.8 ≈
How do the degrees of the monomial terms of
two polynomials affect how they are added or
subtracted? Sample answer: Since only like terms
can be added or subtracted, the degrees of the
monomial terms are important. To be like terms,
they must have the same degree, and they also
must have the same variable with the same
exponent.
About
6154
6154
thousand overseas travelers visited Florida in 2008.
Your Turn
8.
According to the data from the U.S. Census Bureau for 1990–2009, the number of
commercially owned automobiles in the United States can be modeled by the function
A(x) = 1.4x 3 - 130.6x 2 + 1831.3x + 128,141, where x is the number of years after 1990
and A(x) is the number of automobiles in thousands. The number of privately-owned
automobiles in the United States can be modeled by the function
P(x) = -x 3 + 24.9x 2 - 177.9x + 1709.5, where x is the number of years after 1990 and
P(x) is the number of automobiles in thousands. Estimate the total number of automobiles
owned in 2005.
In the equation T(x) = A(x) + P(x), T(x)is the total number of automobiles in thousands.
Add the polynomials.
CONNECT VOCABULARY
(1.4x 3 - 130.6x 2 + 1831.3x + 128141) + (-x 3 + 24.9x 2 - 177.9x + 1709.5)
= (1.4x 3 - x 3) + (-130.6x 2 + 24.9x 2) + (1831.3x - 177.9x) + (128,141 + 1709.5)
To help students remember words and concepts
related to polynomials, have them make a table
similar to the one below showing names, degrees, and
examples of polynomials in standard form up to
degree 5.
= 0.4x 3 - 105.7x 2 + 1653.4x + 129,850.5
The year 2005 is 15 years after 1990, so substitute 15 for x.
0.4(15) - 105.7(15) + 1653.4(15) + 129,850.5 = 132,219
3
2
So, 132,219 thousand automobiles were owned the United States in 2005.
Elaborate
Classifying Polynomials By Degree
Degree
Example
Constant
0
Linear
1
Quadratic
2
⋮
⋮
-9
x+4
2
x + 3x - 1
⋮
SUMMARIZE THE LESSON
What points should you remember when
adding or subtracting polynomials? Sample
answer: combine like terms, write in standard form,
align like terms
9.
© Houghton Mifflin Harcourt Publishing Company
Name
How is the degree of a polynomial related to the degrees of the monomials that comprise the polynomial?
The degree of a polynomial is the degree of the monomial term with the highest degree.
The degree of a monomial is the sum of the exponents of the variables.
10. How is polynomial subtraction based on polynomial addition?
Subtracting two polynomials is the same as adding the first polynomial to the opposite of
the second.
11. How would you find the model for a whole if you have polynomial functions that are models for the two
distinct parts that make up that whole?
To find the function that models the whole, add the two polynomial functions that model
the two distinct parts.
12. Essential Question Check-In What is the result of adding or subtracting polynomials?
The result is always a polynomial.
Module 7
357
Lesson 1
LANGUAGE SUPPORT
A2_MTXESE353930_U3M07L1 357
Connect Vocabulary
Have students work in pairs. Instruct one student in each pair to write a
polynomial of degree 4 in standard form on a large sheet of construction paper.
Both students then write terms from this lesson on separate sticky notes, including
terms such as leading coefficient, monomial, degree, term, first term, coefficient.
Students then decide where to correctly place the sticky notes on the polynomial
expression.
357
Lesson 7.1
22/02/14 8:01 AM
Evaluate: Homework and Practice
1.
EVALUATE
Write the polynomial -23x 7 + x 9 - 6x 3 + 10 + 2x 2 in standard form, and then
identify the degree and leading coefficient.
• Online Homework
• Hints and Help
• Extra Practice
Standard Form: x 9 - 23x 7 - 6x 3 + 2x 2 + 10
Degree: 9
Leading Coefficient: 1
Add the polynomials.
2.
= (82x 8 + 21x 2 - 6) + (7x 8 - 42x 2 + 18x + 3)
Concepts and Skills
Practice
= 89x 8 - 21x 2 + 18x - 3
Explore Activity
Identifying and Analyzing
Monomials and Polynomials
Exercise 1
Example 1
Adding Polynomials
Exercises 2–6
Example 2
Exercises 7–13
= (82x + 7x
8
3.
ASSIGNMENT GUIDE
(82x 8 + 21x 2 - 6) + (18x + 7x 8 - 42x 2 + 3)
) + (21x
8
2
- 42x ) + (18x) + (-6 + 3)
2
(15x - 121x 12 + x 9 - x 7 + 3x 2) + (x 7 - 68x 2 - x 9)
= (-121x 12 + x 9 - x 7 + 3x 2 + 15x) + (-x 9 + x 7 - 68x 2)
= (-121x 12) + (x 9 - x 9) + (-x 7 + x 7) + (3x 2 - 68x 2) + (15x)
= -121x 12 - 65x 2 + 15x
4.
(16 - x 2) + (-18x 2 + 7x 5 - 10x 4 + 5)
Subtracting Polynomials
= (-x 2 + 16) + (7x 5 - 10x 4 - 18x 2 + 5)
= (7x 5) + (-10x 4) + (-x 2 - 18x 2) + (16 + 5)
Example 3
Modeling with Polynomial Addition
and Subtraction
= 7x 5 - 10x 4 - 19x 2 + 21
5.
(x + 1 - 3x ) + (8x + 21x
2
2
- 1)
= (-3x 2 + x + 1) + (21x 2 + 8x - 1)
Exercises 14–20
= (-3x 2 + 21x 2) + (x + 8x) + (1 - 1)
= 18x 2 + 9x
INTEGRATE TECHNOLOGY
(64 + x 3 - 8x 2) + (7x + 3 - x 2) + (19x 2 - 7x - 2)
= (x 3 - 8x 2 + 64) + (-x 2 + 7x + 3) + (19x 2 - 7x - 2)
= (x 3) + (-8x 2 - x 2 + 19x 2) + (7x -7x) + (64 + 3 - 2)
© Houghton Mifflin Harcourt Publishing Company
6.
= x 3 + 10x 2 + 65
7.
(x 4 - 7x 3 + 2 - x) + (2x 3 - 3) + (1 - 5x 3 - x 4 + x)
= (x 4 - 7x 3 - x + 2) + (2x 3 - 3) + (-x 4 - 5x 3 + x + 1)
= (x 4 - x 4) + (-7x 3 + 2x 3 - 5x 3) + (-x + x) + (2 - 3 + 1)
= -10x 3
Subtract the polynomials.
8.
When adding or subtracting two polynomials
in one variable, students can graph the
expressions before they are added (or subtracted),
and after. The graphs should be coincident. The trace
feature in graphing calculators can also be used to
find the certain values for real-world polynomial
functions that are graphed.
(-2x + 23x 5 + 11) - (5 - 9x 3 + x)
= (23x 5 - 2x + 11) + (9x 3 - x - 5)
= (23x 5) + (9x 3) + (-2x - x) + (11 - 5)
= 23x 5 + 9x 3 - 3x + 6
Module 7
Lesson 1
358
Exercise
A2_MTXESE353930_U3M07L1.indd 358
Depth of Knowledge (D.O.K.)
Mathematical Processes
1–12
1 Recall of Information
1.F Analyze relationships
13–20
2 Skills/Concepts
1. A Everyday life
21
2 Skills/Concepts
1.F Analyze relationships
22, 25
3 Strategic Thinking
1.F Analyze relationships
23
3 Strategic Thinking
1.B Problem solving model
24
3 Strategic Thinking
1. A Everyday life
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9.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Communication
(7x 3 + 68x 4 - 14x + 1) - (-10x 3 + 8x + 23)
= (68x 4 + 7x 3 - 14x +1) + (10x 3 - 8x - 23)
= (68x 4) + (7x 3 + 10x 3) + (-14x - 8x) + (1 - 23)
= 68x 4 + 17x 3 - 22x - 22
10. (57x 18 - x 2) - (6x - 71x 3 + 5x 2 + 2)
= (57x 18 - x 2) + (71x 3 - 5x 2 - 6x - 2)
Help students clarify how to add or subtract
polynomials by having them do some of the exercises
in groups. Have one student complete one step of the
addition (or subtraction) process, including an
explanation of the process, then pass the problem to
another student, who completes the second step,
including an explanation. Continue passing the
problem until it is complete.
= (57x 18) + (71x 3) + (-x 2 - 5x 2) + (-6x) + (-2)
= 57x 18 + 71x 3 - 6x 2 - 6x - 2
11. (9x - 12x 3) - (5x 3 + 7x - 2)
= (-12x 3 + 9x) + (-5x 3 - 7x + 2)
= (-12x 3 - 5x 3) + (9x - 7x) + (2)
= -17x 3 + 2x + 2
12. (3x 5 - 9) - (11 + 13x 2 - x 4) - (10x 2 + x 4)
= (3x - 9) + (x 4 - 13x 2 - 11) + (-x 4 - 10x 2)
5
= (3x 5) + (x 4 - x 4) + (-13x 2 - 10x 2) + (-9 - 11)
= 3x 5 - 23x 2 - 20
13.
(10x 2 - x + 4) - (5x + 7) + (6x - 11)
= (10x 2 - x + 4) + (-5x - 7) + (6x - 11)
= (10x 2) + (-x - 5x + 6x) + (4 - 7 - 11)
= 10x 2 - 14
Find the polynomial that models the problem and use it to estimate
the quantity.
© Houghton Mifflin Harcourt Publishing Company
14. A rectangle has a length of x and a width of 5x 3 + 4 - x 2. Find the perimeter of the
rectangle when the length is 5 feet.
P(x) = ℓ(x) + ℓ(x) + w(x) + w(x)
= x + x + (5x 3 + 4 - x 2) + (5x 3 + 4 - x 2)
= (5x 3 + 5x 3) + (-x 2 - x 2) + (x + x) + (4 + 4)
= 10x 3 - 2x 2 + 2x + 8
P(5) = 10(5) - 2(5) + 2(5) + 8 = 1218
3
The perimeter is 1218 feet.
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15. A rectangle has a perimeter of 6x 3 + 9x 2 - 10x + 5 and a length of x. Find the width of the rectangle when
the length is 21 inches.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Math Connections
w(x) + w(x) = P(x) - ℓ(x) - ℓ(x)
= (6x 3 + 9x 2 - 10x + 5) -x - x
= 6x 3 + 9x 2 + (-10x - x - x) + 5
Remind students that they have previously worked
with polynomial functions and they should build on
that knowledge. Have students write a quartic
polynomial function f in standard form and identify
the leading coefficient and the number of terms.
Then have them graph f, describe the graph, and give
other attributes of the polynomial, including the
expected end behavior.
= 6x 3 + 9x 2 - 12x + 5
2w(21) = 6(21) + 9(21) - 12(21) + 5 = 59,288
3
2
Since 59,288 is twice the width, the width of the rectangle is half of this, or 29,644.
The width is 29,644 inches when the length is 21 inches.
16. Cho is making a garden, where the length is x feet and the
width is 4x - 1 feet. He wants to add garden stones around the
perimeter of the garden once he is done. If the garden is 4 feet
long, how many feet will Cho need to cover with garden stones?
P(x) = ℓ(x) + ℓ(x) + w(x) + w(x)
= x + x + (4x - 1) + (4x - 1)
= (x + x + 4x + 4x) + (-1 -1)
= 10x - 2
P(4) = 10(4) -2 = 38
The perimeter is 38 feet when the length is 4 feet long, so Cho will need to
cover 38 feet with garden stones.
In the equation F(x) = T(x) - M(x), F(x) is the median weekly earnings in dollars.
(0.012x 3 - 0.46x 2 + 56.1x + 732.3) - (0.009x 3 - 0.29x 2 + 30.7x + 439.6)
= (0.012x 3 - 0.46x 2 + 56.1x + 732.3) + (-0.009x 3 + 0.29x 2 - 30.7x - 439.6)
= (0.012x 3 - 0.009x 3) + (-0.46x 2 - 0.29x 2) + (56.1x - 30.7x) + (732.3 - 439.6)
= 0.003x 3 - 0.17x 2 + 25.4x + 292.7
The year 2010 is 30 years after 1980, so substitute 30 for x.
0.003(30) - 0.17(30) + 25.4(30) + 292.7 ≈ 983
3
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© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Edward
Bock/Corbis
17. Employment The data from the U.S. Census Bureau for 1980–2010 shows that the median weekly
earnings of full-time male employees who have at least a bachelor’s degree can be modeled by the function
M(x) = 0.009x 3 - 0.29x 2 + 30.7x + 439.6, where x is the number of years after 1980 and M(x) is the
median weekly earnings in dollars. The median weekly earnings of all full-time employees who have at
least a bachelor’s degree can be modeled by the function T(x) = 0.012x 3 - 0.46x 2 + 56.1x + 732.3, where
x is the number of years after 1980 and T(x) is the median weekly earnings in dollars. Estimate the median
weekly earnings of a full-time female employee with at least a bachelor’s degree in 2010.
Lesson 1
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18. Business From data gathered in the period 2008–2012, the yearly value of U.S. exports can be modeled
by the function E(x) = -228x 3 + 2552.8x 2 - 6098.5x + 11,425.8, where x is the number of years after
2008 and E(x) is the value of exports in billions of dollars. The yearly value of U.S. imports can be modeled
by the function l(x) = -400.4x 3 + 3954.4x 2 - 11,128.8x + 17,749.6, where x is the number of years
after 2008 and l(x) is the value of imports in billions of dollars. Estimate the total value the United States
imported and exported in 2012.
AVOID COMMON ERRORS
Students may identify the wrong term as the leading
coefficient or find the wrong degree of a polynomial.
They may benefit from rewriting polynomials in
standard form before identifying the degree and
leading coefficient.
In the equation T(x) = E(x) + l(x), T(x) is the total value of imports and exports in billions of
dollars.
(-228x 3 + 2252.8x 2 - 6098.5x + 11,425.8) + (-400.4x 3 + 3954.4x 2 - 11,128.8x + 17,749.6)
= (-228x 3 - 400.4x 3) + (2252.8x 2 + 3954.4x 2) + (-6098.5x - 11,128.8x) + (11,425.8 + 17,749.6)
= -628.4x 3 + 6207.2x 2 - 17,227.3x + 29,175.4
The year 2008 is 4 years after 2012, so substitute 4 for x.
-628.4(4) + 6207.2(4) - 17,227.3(4) + 29,175.4 ≈ 19,364
3
2
19. Education From data gathered in the period
1970–2010, the number of full-time students enrolled in a
degree-granting institution can be modeled by the function
F(x) = 8.7x 3 - 213.3x 2 + 2015.5x + 3874.9, where x is the
number of years after 1970 and F(x) is the number of students
in thousands. The number of part-time students enrolled in
a degree-granting institution can be modeled by the function
P(x) = 12x 3 - 285.3x 2 + 2217x + 1230, where x is the number of
years after 1970 and P(x) is the number of students in thousands.
Estimate the total number of students enrolled in a degree-granting
institution in 2000.
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©MarmadukeSt. John/Alamy
In the equation T(x) = F(x) + P(x), T(x) is the total number of students in thousands.
(8.7x 3 - 213.3x 2 + 2015.5 + 3874.9) + (12x 3 - 285.3x 2 + 2217x + 1230)
=( 8.7x 3 + 12x 3) + (-213.3x 2 - 285.3x 2) + (2015.5x + 2217x) + (3874.9 + 1230)
= 20.7x 3 - 498.6x 2 + 4232.5x + 5104.9
The year 2000 is 30 years after 1970, so substitute 30 for x.
20.7(30) - 498.6(30) + 4232.5(30) + 5104.9 ≈ 242,240
3
About 242,240 thousand students were enrolled in 2000.
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20. Geography The data from the U.S. Census Bureau for 1982–2003 shows that the surface
area of the United States that is covered by rural land can be modeled by the function
R(x) = 0.003x 3 - 0.086x 2 - 1.2x + 1417.4, where x is the number of years after 1982 and R(x) is
the surface area in millions of acres. The total surface area of the United States can be modeled by
the function T(x) = 0.0023x 3 + 0.034x 2 - 5.9x + 1839.4, where x is the number of years after
1982 and T(x) is the surface area in millions of acres. Estimate the surface area of the United States
that is not covered by rural land in 2001.
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Communication
When students work with polynomials and their
exponents, they might confuse coefficients and
exponents. Have students read problems aloud,
stating the terms clearly and carefully–for example,
either “four x” or “x to the fourth”—as appropriate.
In the equation N(x) = T(x) - R(x), N(x) is the surface area in millions of acres that is not
covered by rural land.
(0.0023x 3 + 0.034x 2 - 5.9x + 1839.4) - (0.003x 3 - 0.086x 2 - 1.2x + 1417.4)
= (0.0023x 3 + 0.034x 2 - 5.9x + 1839.4) + (-0.003x 3 + 0.086x 2 + 1.2x - 1417.4)
= (0.0023x 3 - 0.003x 3) + (0.034x 2 + 0.086x 2) + (-5.9x + 1.2x) + (1839.4 - 1417.4)
= -0.0007x 3 + 0.12x 2 - 4.7x + 422
N(19) = -0.0007(19) + 0.12(19) - 4.7(19) + 422 ≈ 371
3
2
AVOID COMMON ERRORS
About 371 millions of acres were not covered by rural land in 2001.
Students may make sign errors when subtracting
polynomials. Point out that rewriting the subtraction
of polynomials as an addition of the opposite may
prevent errors.
21. Determine which polynomials are monomials. Choose all that apply.
a. 4x 3y
e. x
b. 12 - x 2 + 5x
f.
c.
152 + x
19x
-2
4 2
g. 4x x
d. 783
H.O.T. Focus on Higher Order Thinking
22. Explain the Error Colin simplified (16x + 8x 2y - 7xy 2 + 9y - 2xy) - (-9xy + 8xy 2 + 10x 2y + x - 7y).
His work is shown below. Find and correct Colin’s mistake.
(16x + 8x 2y - 7xy 2 + 9y - 2xy) - (-9xy + 8xy 2 + 10x 2y + x - 7y)
= (16x + 8x 2y - 7xy 2 + 9y - 2xy) + (9xy - 8xy 2 - 10x 2y - x + 7y)
= (16x - x) + (8x 2y - 7xy 2 - 8xy 2 - 10x 2y) + (9y + 7y) + (-2xy + 9xy)
© Houghton Mifflin Harcourt Publishing Company
= 15x - 17x 2y 2 + 16y + 7xy
Colin incorrectly combined the terms 8x 2y, -7xy 2 , 8xy 2 and 10x y. The like terms are
2
8x 2y and 10x y, and -7xy 2 and 8xy 2.
2
(16x + 8x 2y - 7xy 2 + 9y - 2xy) - (-9xy + 8xy 2 + 10x 2y + x - 7y)
= (16x + 8x 2y - 7xy 2 + 9y - 2xy) + (9xy - 8xy 2 - 10x 2y - x + 7y)
= (16x - x) + (8x 2y - 10x 2y) + (-7xy 2 - 8xy 2 ) + (9y + 7y) + (-2xy + 9xy)
= 15x - 2x 2y - 15xy 2 + 16y + 7xy
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23. Critical Reasoning Janice is building a fence around a
portion of her rectangular yard. The length of yard she will
enclose is x, and the width is 2x 2 − 98x + 5, where the
measurements are in feet. If the length of the enclosed yard
is 50 feet and the cost of fencing is $13 per foot, how much
will Janice need to spend on fencing?
PEERTOPEER DISCUSSION
Instruct one student in each pair to write two
polynomials in standard form while the other student
gives verbal instructions for adding the polynomials.
Then have students switch roles and repeat the
exercise, giving instructions for subtracting the
polynomials.
In the equation P(x) = l(x) + l(x) + w(x) + w(x),
P(x) is the perimeter of the enclosed yard in feet.
x + x + (2x 2 - 98x + 5) + (2x 2 - 98x + 5)
= (2x 2 + 2x 2 ) + (x + x - 98x - 98x) + (5 + 5)
= 4x 2 - 194x + 10
P(50) = 4(50) 2 - 194(50) + 10 = 310
JOURNAL
The perimeter of the yard is 310 feet.
$13 ⋅ 310 = $4030
Have students write about how polynomials are
related to monomials, and how the degrees of
monomials are important when adding or
subtracting polynomials.
Janice will need to spend $4030 on fencing.
24. Multi-Step Find a polynomial expression for the perimeter of a trapezoid with legs of length x and bases
of lengths 7x 3 + 2x and x 2 + 3x - 10 where each is measured in inches.
a. Find the perimeter of the trapezoid if the length of one leg is 6 inches.
b. If the length is increased by 5 inches, will the perimeter also increase?
By how much?
P(x) = x + x + (7x 3 + 2x) + (x 2 + 3x - 10)
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©berna
namoglu/Shutterstock
= (7x 3) + (x 2) + (x + x + 2x + 3x) + (-10)
= 7x 3 + x 2 + 7x - 10
3
2
a. 7(6) + (6) + 7(6) − 10 = 1580
The perimeter is 1580 inches.
3
2
b. 7(11) + (11) + 7(11) − 10 = 9505
9505 − 1580 = 7925
The perimeter will increase by 7925 inches.
25. Communicate Mathematical Ideas Present a formal argument for why the set of polynomials is closed
under addition and subtraction. Use the polynomials ax m + bx m and ax m - bx m, for real numbers a and b
and whole number m, to justify your reasoning.
ax m + bx m = (a + b)x m
The sum of two monomials, ax m and bx m, is another monomial, (a + b)x m.
ax m − bx m = (a - b)x m
The difference of two monomials, ax m and bx m, is another monomial, (a - b)x m
So, the sum or difference of two polynomials is a sum of monomials, which is another
polynomial.
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Lesson Performance Task
CONNECT CONTEXT
Some students may not be familiar with the name
and location of Death Valley. Point out the location
on a map, and explain how the high temperatures
relate to the name Death Valley.
The table shows the average monthly maximum and minimum temperatures for Death Valley
throughout one year.
Month
Maximum Temperature
Minimum Temperature
January
67
40
February
73
46
March
82
55
AVOID COMMON ERRORS
April
91
62
May
101
73
June
110
81
July
116
88
August
115
86
Some students may introduce sign errors when doing
subtraction. When two functions are subtracted, all
terms in the second set of parentheses should change
sign: (a + b) - (-c + d) = a + b + c - d.
September
107
76
October
93
62
November
77
48
December
65
38
INTEGRATE MATHEMATICAL
PROCESSES
Focus on Critical Thinking
Use a graphing calculator to find a good fourth-degree polynomial regression model for both
the maximum and minimum temperatures. Then find a function that models the range in
monthly temperatures and use the model to estimate the range during September. How does
the range predicted by your model compare with the range shown in the table?
Ask students how they might improve the accuracy of
their models. Have them discuss the advantages of
adding more data points versus using a higher-order
polynomial.
Maximum temperature model:
T max(t) = 0.0319t 4 - 0.962t 3 + 8.12t 2 - 15.2t + 76.2
T min(t) = 0.0372t 4 - 1.08t 3 + 8.97t 2 - 17.8t + 51.4
where t is time (1 = Jan., 2 = Feb., etc.) and T is temperature (degrees Fahrenheit).
The range of monthly temperatures is the difference of the maximum and minimum
temperatures.
R(t) = T max(t) - T min(t)
R(t) = (0.0319t 4 - 0.962t 3 + 8.12t 2 - 15.2t + 76.2) - (0.0372t 4 - 1.08t 3 + 8.97t 2 - 17.8t + 51.4)
R(t) = (0.0319t 4 - 0.0372t 4) + (-0.962t 3 + 1.08t 3) + (8.12t 2 - 8.97t 2) +
(-15.2t + 17.8t) + (76.2 - 51.4)
R(t) = -0.0053t 4 + 0.118t 3 - 0.85t 2 + 2.6t + 24.8
© Houghton Mifflin Harcourt Publishing Company
Minimum temperature model:
R(9) = -0.0053(9) 4 + 0.118(9) 3 - 0.85(9) + 2.6(9) + 24.8
2
R(9) = -0.0053(6561) + 0.118(729) - 0.85(81) + 2.6(9) + 24.8
R(9) = -34.8 + 86.0 - 68.9 + 23.4 + 24.8 = 30.5
The predicted range in temperatures in September is about 30.5 degrees, which is in agreement
with the September temperatures shown in the data table.
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Have the students use their maximum and minimum temperature models to
calculate the average temperature for the month of September. Then have students
research the actual average temperature in Death Valley for September. Ask
students whether their models accurately predicted the average temperature and,
if not, what might account for the difference. For example, the data might be
based on different time periods, and so a September model would not be
accurate.
21/02/14 7:04 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.
Adding and Subtracting Polynomials
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