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Name Class Date
Explore Identifying and Analyzing Monomials
and Polynomials
A polynomial function of degree n has the standard form p(x) = anxn^ + an - 1 xn -^1 + … + a 2 x^2 + a 1 x + a 0 , where an, an - 1 ,…, a 2 , a 1 , and a 0 are real numbers and a n ≠ 0. The expression anxn^ + an - 1 x n -^1 + … a 2 x^2 + a 1 x + 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.
A Identify the monomials:^ x
(^3) , y + 3 y (^2) - 5 y (^3) + 10, a (^2) bc (^12) , 76
Monomials:
Not monomials:
B Identify the degree of each monomial.
C Identify the terms of the polynomial^ y^ +^^3 y
(^2) - 5 y (^3) + 10.
D Identify the coefficient of each term.
E Identify the degree of each term.
F Write the polynomial in standard form.
G What is the leading coefficient of the polynomial?
Monomial x^3 a^2 bc^12
Degree
Term y 3 y^2 - 5 y^3 Coefficient
Term y 3 y^2 - 5 y^3
Degree
Resource Locker
x^3 , a^2 bc^12 , 76
y + 3 y^2 - 5 y^3 + 10
y , 3 y^2 , - 5 y^3 , 10
- 5 y^3 + 3 y^2 + y + 10
Module 6 315 Lesson 1
6.1 Adding and Subtracting
Polynomials
Essential Question: How do you add or subtract two polynomials, and what type of expression is the result?
IN3_MNLESE389885_U3M06L1 315 6/10/15 1:16 PM
Common Core Math Standards
The student is expected to:
A-APR.
Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of addition, subtraction,
and multiplication; add, subtract, and multiply polynomials. Also F.BF.1b
Mathematical Practices
MP.2 Reasoning
Language Objective
Students work in pairs to create a “parts of a polynomial” chart.
HARDBOUND SE
PAGE 249
BEGINS HERE
Turn to these pages to
find this lesson in the
hardcover student
edition.
HARDCOVER PAGES 249
Adding and
Subtracting
Polynomials
ENGAGE
Essential Question: How do you add
or subtract two polynomials, and what
type of expression is the result?
First, add or subtract like terms. The sum or
difference is another polynomial.
PREVIEW: LESSON
PERFORMANCE TASK
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.
315
Turn to these pages to
find this lesson in the
hardcover student
edition.
HARDCOVER
© Houghton Mifflin Harcourt Publishing Company
Name Class
Date
Explore^ Identifying and Analyzing Monomialsand Polynomials A polynomial function of degree^ n^ has the^ standard form
p ( x ) = an xn^ +^ an -^1 xn -^1 +^ …^ +^ a^2 x^2 + a 1 x +^ a^0 , where^ a^ n,^ a^ n -^1 ,…,^ a^2 ,^ a^1 , and^ a^0 are real numbers and
a^ n ≠^ 0. The expression^ a^ n^ x^ n^ + an - 1 x^ n -^1 +^ …^ a^2 x^2 +^ a^1 x^ +^ a^0 is called a^ polynomial
, and each term of a polynomial is called a monomial exponents. A polynomial is a monomial or a sum of monomials. The. A monomial is the product of a number and one or more variables with whole-number^ 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. Thegreatest degree.^ leading coefficient^ of a polynomial is the coefficient of the term with the Identify the monomials:^ x^3 ,^ y^ +^^3 y^
(^2) - 5 y 3 + 10, a^2 bc^12 , 76 Monomials:Not monomials: Identify the degree ofeach monomial. Identify the terms of the polynomial^ y^ +
3 y 2 - 5 y 3 +^ 10. ^ Identify the coefficientof each term. ^ Identify the degree ofeach term. Write the polynomial in standard form. What is the leading coefficient of the polynomial?
Monomial^ x^3 a^2 bc^12
76 Degree Term Coefficient y^3 y^2 -^5 y^3 Term Degree y^3 y^2 -^5 y^3
ResourceLocker A-APR.1^ For the full text of this standard, see the table starting on page CA2. Also F.BF.1b
x^3 y, a +^2 bc 3 y^122 , 76- 5 y (^3) + 10 (^3) y , 3y^152 , - 5 y (^3) , 10^0 1 3 -^510
-^15 y^3 +^3 y^2 +^2 y^ +^103 - 5
Module 6
315 Lesson 1
6. 1^ Adding and SubtractingPolynomials Essential Question:^ How do you add or subtract two polynomials, and what type ofexpression is the result?
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-C;CA-C
IN3_MNLESE389885_U3M06L1^315
09/06/15^ 11:38 AM
315 Lesson 6. 1
L E S S O N
© Houghton Mifflin Harcourt Publishing Company
Reflect
- Discussion How can you find the degree of a polynomial with multiple variables in each term?
Explain 1 Adding Polynomials
To add polynomials, combine like terms.
Example 1 Add the polynomials.
(^4 x
2 - x 3 + 2 + 5 x 4 ) + (-x + 6 x 2 + 3 x 4 )
5 x^4 - x^3 + 4 x^2 + 2 Write in standard form. ___^ +^3 x^4 +^6 x^2 - x 8 x^4 - x^3 + 10 x^2 - x + 2
Align like terms.
(^10 x^ -^18 x
3 + 6 x 4 - 2 ) + (- 7 x 4 + 5 + x + 2 x 3 )
( 6 x^4 - 18 x^3 + 10 x - 2 ) + (- 7 x^4 + 2 x^3 + x + 5 ) Write in standard form.
= (^) ( 6 x^4 - (^) ) + (^) ( + 2 x^3 ) + (^) ( + x) + (^) (- 2 + (^) ) Group like terms.
= - 16 x^3 + + 3 Add.
Reflect
- Is the sum of two polynomials always a polynomial? Explain.
Your Turn
Add the polynomials.
3. ( 17 x^4 + 8 x^2 - 9 x^7 + 4 - 2 x^3 ) + ( 11 x^3 - 8 x^2 + 12 )
4. (- 8 x + 3 x^11 + x^6 ) + ( 4 x^4 - x + 17 )
Add.
Find the degree of each term by adding the exponents of each variable. The degree of the
polynomial is the degree of the term with the highest degree.
7 x^4 10 x
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.
(- 9 x^7 + 17 x^4 - 2 x^3 + 8 x^2 + 4 ) + ( 11 x^3 - 8 x^2 + 12 )
= (- 9 x^7 ) +( 17 x^4 ) + (- 2 x^3 + 11 x^3 ) + (- 8 x^2 - 8 x^2 ) + ( 4 + 12 )
= - 9 x^7 + 17 x^4 + 9 x^3 + 16
( 3 x^11 + x^6 - 8 x ) + ( 4 x^4 - x + 17 )
= ( 3 x^11^ )^ + ( x^6^ )^ + ( 4 x^4^ )^ + (- 8 x - x ) + ( 17 )
= 3 x^11 + x^6 + 4 x^4 - 9 x + 17
Module 6 316 Lesson 1
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HARDBOUND SE
PAGE 250
BEGINS HERE EXPLORE
Identifying and Analyzing Monomials
and Polynomials
INTEGRATE TECHNOLOGY
Students have the option of completing the
polynomial activity either in the book or
online.
QUESTIONING STRATEGIES
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.
How do you recognize the leading
coefficient? Write the polynomial in standard
form. It is the coefficient of the term with the
highest degree.
EXPLAIN 1
Adding Polynomials
AVOID COMMON ERRORS
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.
PROFESSIONAL DEVELOPMENT
Learning Progressions
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 (^) n x n^ + a (^) n - 1 x n^ -^^1 + … + a 2 x^2 + a 1 x + 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 ai 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 study in future courses.
Adding and Subtracting Polynomials 316
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jutta
Klee/Corbis
Explain 3 Modeling with Polynomial Addition and
Subtraction
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 + 141 x + 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.
( - 10.4 x^3 + 74.2 x^2 - 3.4x + 8320.2) + ( - 13.8x^3 + 55.3x^2 + 141 x + 7880 )
= ( - 10.4 x^3 - 13.8 x^3 )^ + (74.2x^2 + 55.3x^2 ) + ( - 3.4x + 141 x ) + ( 8320.2 + 7880 )
= - 24.2 x^3 + 129.5 x^2 + 137.6x + 16,200.
The year 2009 is 4 years after 2005, so substitute 4 for x.
- 24.2( 4 )^3 + 129.5( 4 )^2 + 137.6( 4 ) + 16,200.2 ≈ 17,
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– shows that the total number of overseas travelers visiting New York and Florida can be modeled by the function T (x) = 41.5 x^3 - 689.1 x^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) = - 41.6 x^3 + 560.9 x^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.
Subtract the polynomials.
(41.5 x^3 - 689.1 x^2 + 4323.3x + 2796.6) ( - 41.6x^3 + 560.9x^2 - 1632.7x + 6837.4)
= (41.5 x^3 - 689.1 x^2 + 4323.3x + 2796.6) + (41.6x^3 - 560.9x^2 + 1632.7x - 6837.4)
Module 6 318 Lesson 1
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HARDBOUND SE
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BEGINS HERE
DIFFERENTIATE INSTRUCTION
Visual Cues
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.
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.
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.
EXPLAIN 3
Modeling with Polynomial Addition
and Subtraction
INTEGRATE MATHEMATICAL PRACTICES
Focus on Modeling
MP.4 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.
QUESTIONING STRATEGIES
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.
Adding and Subtracting Polynomials 318
© Houghton Mifflin Harcourt Publishing Company
= (^) (41.5x^3 + (^) ) + (^) ( - 560.9x^2 ) + (^) ( + 1632.7x) + (^) (2796.6 - (^) )
= x^3 - x^2 + x -
The year 2008 is 8 years after 2000, so substitute for x.
83.1( 8 )^3 - 1250 ( 8 )^2 + 5956 ( 8 ) - 4040.8 ≈
About thousand overseas travelers visited Florida in 2008.
Your Turn
- 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) = 1.4 x^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.
Elaborate
- How is the degree of a polynomial related to the degrees of the monomials that comprise the polynomial?
- How is polynomial subtraction based on polynomial addition?
- 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?
- Essential Question Check-In What is the result of adding or subtracting polynomials?
- 689.1x^2
In the equation T ( x ) = A ( x ) + P ( x ) , T ( x ) is the total number of automobiles in thousands. Add the polynomials.
(1.4 x^3 - 130.6x^2 + 1831.3x + 128141 ) + (- x^3 + 24.9x^2 - 177.9x + 1709.5)
= (1.4 x^3 - x^3 )^ + (-130.6x^2 + 24.9x^2 ) + (1831.3x - 177.9x )^ + (128,141 + 1709.5)
= 0.4x^3 - 105.7x^2 + 1653.4x + 129,850. The year 2005 is 15 years after 1990, so substitute 15 for x. 0.4( 15 )^3 - 105.7( 15 )^2 + 1653.4( 15 ) + 129,850.5 = 132, So, 132,219 thousand automobiles were owned in the United States in 2005.
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.
Subtracting two polynomials is the same as adding the first polynomial to the opposite of
the second.
To find the function that models the whole, add the two polynomial functions that model
the two distinct parts.
The result is always a polynomial.
41.6 x^3 4323.3x 6837.
Module 6 319 Lesson 1
IN3_MNLESE389885_U3M06L1 319 09/06/15 11:35 AM
LANGUAGE SUPPORT
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.
ELABORATE
QUESTIONING STRATEGIES
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.
CONNECT VOCABULARY
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.
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
Classifying Polynomials By Degree
Name Degree Example
Constant 0 - 9
Linear 1 x + 4
Quadratic 2 x^2 + 3 x - 1
319 Lesson 6. 1
© Houghton Mifflin Harcourt Publishing Company
9. ( 7 x^3 + 68 x^4 - 14 x + 1 ) - ( - 10 x^3 + 8 x + 23 )
10. ( 57 x^18 - x^2 ) - ( 6 x - 71 x^3 + 5 x^2 + 2 )
11. ( 9 x - 12 x^3 )^ - ( 5 x^3 + 7 x - 2 )
12. ( 3 x^5 - 9 ) - ( 11 + 13 x^2 - x^4 ) - ( 10 x^2 + x^4 )
13. ( 10 x^2 - x + 4 ) - ( 5 x + 7 ) + ( 6 x - 11 )
Find the polynomial that models the problem and use it to estimate the quantity.
- 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.
= ( 68 x^4 + 7 x^3 - 14 x + 1 ) + ( 10 x^3 - 8 x - 23 )
= ( 68 x^4 )^ + ( 7 x^3 + 10 x^3 ) + (- 14 x - 8 x )^ + ( 1 - 23 )
= 68 x^4 + 17 x^3 - 22 x - 22
= ( 57 x^18 - x^2 )^ + ( 71 x^3 + 5 x^2 - 6 x - 2 )
= ( 57 x^18 )^ + ( 71 x^3 )^ + (- x^2 - 5 x^2 )^ + (- 6 x ) + (- 2 )
= 57 x^18 + 71 x^3 - 6 x^2 - 6 x - 2
= (- 12 x^3 + 9 x ) + (- 5 x^3 - 7 x + 2 )
= (- 12 x^3 - 5 x^3 )^ + ( 9 x - 7 x )^ + ( 2 )
= - 17 x^3 + 2 x + 2
= ( 3 x^5 - 9 )^ + (x^4 - 13 x^2 - 11 )^ + (- x^4 - 10 x^2 )
= ( 3 x^5 ) + (x^4 - x^4 ) + (- 13 x^2 - 10 x^2 ) + (- 9 - 11 )
= 3 x^5 - 23 x^2 - 20
= ( 10 x^2 - x + 4 ) + (- 5 x - 7 ) + ( 6 x - 11 )
= ( 10 x^2 ) + (-x - 5 x + 6 x ) + ( 4 - 7 - 11 )
= 10 x^2 - 14
P ( x )^ = ℓ ( x )^ + ℓ ( x )^ + w ( x )^ + w ( x )
= x + x + ( 5 x^3 + 4 - x^2 ) + ( 5 x^3 + 4 - x^2 )
= ( 5 x^3 + 5 x^3 ) + (- x^2 - x^2 )^ + (x + x ) + ( 4 + 4 )
= 10 x^3 - 2 x^2 + 2 x + 8 P ( 5 )^ = 10 ( 5 )
3
- 2 ( 5 )
**2
- 2 ( 5 )**^ + 8 = 1218 The perimeter is 1218 feet.
Module 6 321 Lesson 1
IN3_MNLESE389885_U3M06L1.indd 321 4/7/14 2:42 PM
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 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.
321 Lesson 6. 1
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Edward
Bock/Corbis
- A rectangle has a perimeter of 6x^3 + 9 x^2 - 10 x + 5 and a length of x. Find the width of the rectangle when the length is 21 inches.
- Cho is making a rectangular 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?
- 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) = 0.009 x^3 - 0.29 x^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) = 0.012 x^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.
P ( x ) = ℓ ( x ) + ℓ ( x ) + w ( x ) + w ( x ) = x + x + ( 4 x - 1 )^ + ( 4 x - 1 )
= (x + x + 4 x + 4 x ) + (- 1 - 1 )
= 10 x - 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.
(0.012 x^3 - 0.46x^2 + 56.1x + 732.3) - (0.009 x^3 - 0.29 x^2 + 30.7x + 439.6)
= (0.012 x^3 - 0.46x^2 + 56.1x + 732.3) + (-0.009 x^3 + 0.29 x^2 - 30.7x - 439.6)
= (0.012 x^3 - 0.009x^3 )^ + (-0.46 x^2 + 0.29 x^2 )^ + (56.1x - 30.7x )^ + (732.3 - 439.6)
= 0.003x^3 - 0.17x^2 + 25.4x + 292. The year 2010 is 30 years after 1980, so substitute 30 for x.
3
- 0.17( 30 )
2
+ 25.4( 30 )^ + 292.7 ≈ 983 dollars
w ( x ) + w ( x ) = P ( x ) - ℓ ( x ) - ℓ ( x )
= ( 6 x^3 + 9 x^2 - 10 x + 5 ) - x - x
= 6 x^3 + 9 x^2 + (- 10 x - x - x )^ + 5
= 6 x^3 + 9 x^2 - 12 x + 5 2 w ( 21 ) = 6 ( 21 )^3 + 9 ( 21 )^2 - 12 ( 21 ) + 5 = 59, 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.
Module 6 322 Lesson 1
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HARDBOUND SE
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BEGINS HERE
INTEGRATE MATHEMATICAL PRACTICES
Focus on Math Connections
MP.1 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.
Adding and Subtracting Polynomials 322
© Houghton Mifflin Harcourt Publishing Company
- 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) = 0.003 x^3 - 0.086 x^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.
- Determine which polynomials are monomials. Choose all that apply. a. 4 x^3 y
b. 12 - x^2 + 5 x
c. 152 + x d. 783
e. x
f. 19 x -^2
g. 4 x^4 x^2
H.O.T. Focus on Higher Order Thinking
- Explain the Error Colin simplified ( 16 x + 8 x^2 y - 7 xy^2 + 9 y - 2 xy) - ( - 9 xy + 8 xy^2 + 10 x^2 y + x - 7 y). His work is shown below. Find and correct Colin’s mistake. (^16 x^ +^^8 x^2 y^ -^^7 x y^2 +^^9 y^ -^^2 xy ) -^ ( -^9 xy^ +^^8 x y^2 +^^10 x^2 y^ +^ x^ -^^7 y ) = (^) ( 16 x + 8 x^2 y - 7 x y^2 + 9 y - 2 xy ) + (^) ( 9 xy - 8 x y^2 - 10 x^2 y - x + 7 y ) = ( 16 x - x) + (^) ( 8 x^2 y - 7 xy^2 - 8 x y^2 - 10 x^2 y) + (^) ( 9 y + 7 y) + (^) ( - 2 xy + 9 xy) = 15 x - 17 x^2 y^2 + 16 y + 7 xy
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.
(0.0023 x^3 + 0.034x^2 - 5.9x + 1839.4) - (0.003 x^3 - 0.086 x^2 - 1.2x + 1417.4)
= (0.0023x^3 + 0.034x^2 - 5.9x + 1839.4) + (-0.003 x^3 + 0.086 x^2 + 1.2x - 1417.4)
= (0.0023x^3 - 0.003x^3 ) + ( 0.034 x^2 + 0.086 x^2 ) + (-5.9x + 1.2x ) + (1839.4 - 1417.4)
= - 0.0007x^3 + 0.12x^2 - 4.7x + 422
N( 19 )^ = - 0.0007( 19 )^3 + 0.12( 19 )^2 - 4.7( 19 )^ + 422 ≈ 371
About 371 million acres were not covered by rural land in 2001.
Colin incorrectly combined the terms 8x^2 y, - 7 xy^2 , - 8 xy 2 and - 10 x^2 y. The like terms are 8 x^2 y and 10 x
2 y, and - 7 xy^2 and 8xy 2.
( 16 x + 8 x^2 y - 7 xy^2 + 9 y - 2 xy ) - (- 9 xy + 8 xy^2 + 10 x^2 y + x - 7 y )
= (^) ( 16 x + 8 x 2 y - 7 xy 2 + 9 y - 2 xy ) + (^) ( 9 xy - 8 xy 2 - 10 x 2 y - x + 7 y ) = ( 16 x - x ) + (^) ( 8 x^2 y - 10 x^2 y ) + (^) (- 7 xy 2 - 8 xy (^2) ) + ( 9 y + 7 y ) + (^) (- 2 xy + 9 xy ) = 15 x - 2 x 2 y - 15 xy 2 + 16 y + 7 xy
Module 6 324 Lesson 1
IN3_MNLESE389885_U3M06L1.indd 324 2/29/16 9:44 PM
HARDBOUND SE
PAGE 255
BEGINS HERE
INTEGRATE MATHEMATICAL PRACTICES
Focus on Communication
MP.3 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.
AVOID COMMON ERRORS
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.
Adding and Subtracting Polynomials 324
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©bernanamoglu/Shutterstock
- 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 − 98 x + 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?
- Multi-Step Find a polynomial expression for the perimeter of a trapezoid with legs of length x and bases of lengths 0.1x^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 leg length is increased by 5 inches, will the perimeter also increase? By how much?
- Communicate Mathematical Ideas Present a formal argument for why the set of polynomials is closed under addition and subtraction. Use the polynomials axm^ + bxm^ and axm^ - bxm , for real numbers a and b and whole number m, to justify your reasoning.
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 + ( 2 x^2 - 98 x + 5 ) + ( 2 x^2 - 98 x + 5 )
= ( 2 x^2 + 2 x^2 ) + ( x + x - 98 x - 98 x ) + ( 5 + 5 )
= 4 x^2 - 194 x + 10 P ( 50 )^ = 4 ( 50 )^2 - 194 ( 50 )^ + 10 = 310 The perimeter of the yard is 310 feet. $13 ⋅ 310 = $ Janice will need to spend $4030 on fencing.
P ( x ) = x + x + (0.1 x^3 + 2 x ) + (x^2 + 3 x - 10 )
= (0.1 x^3 )^ + ( x^2 )^ + ( x + x + 2 x + 3 x ) + ( - 10 )
= 0.1x^3 + x 2 + 7 x - 10
a. 0.1( 6 )
3
2
+ 7 ( 6 )^ − 10 = 89.
The perimeter is 89.6 inches.
b. 0.1( 11 )^3 + ( 11 )^2 + 7 ( 11 ) − 10 = 321.
The perimeter will increase by 231.5 inches.
a x m^ + bx m^ = (a + b ) xm
The sum of two monomials, ax m^ and bx m, is another monomial, (a + b ) xm^.
a x m^ − bx m^ = (a - b ) xm
The difference of two monomials, ax m^ and bx m, is another monomial, (a - b ) xm
So, the sum or difference of two polynomials is a sum of monomials, which is another polynomial.
Module 6 325 Lesson 1
IN3_MNLESE389885_U3M06L1 325 09/06/15 11:36 AM
PEERTOPEER 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.
JOURNAL
Have students write about how polynomials are
related to monomials, and how the degrees of
monomials are important when adding or
subtracting polynomials.
325 Lesson 6. 1
