Name
Class
6.1
Date
Adding and Subtracting Polynomials
Essential Question: How do you add or subtract two polynomials, and what type of expression is the result?
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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 and a n ≠ 0. 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.
A
Identify the monomials: x 3, y + 3y 2 - 5y 3 + 10, a 2 bc 12, 76 Monomials: Not monomials:
B
Identify the degree of each monomial.
Monomial
x3
a 2 bc 12
76
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Degree
C
Identify the terms of the polynomial y + 3y 2 - 5y 3 + 10.
D
Identify the coefficient of each term.
E
Identify the degree of each term.
Term
3y 2
y
-5y 3
10
-5y 3
10
Coefficient
Term
3y 2
y
Degree
F
Write the polynomial in standard form.
G
What is the leading coefficient of the polynomial?
Module 6
315
Lesson 1
Reflect
Discussion How can you find the degree of a polynomial with multiple variables in each term?
1.
Explain 1
Adding Polynomials
To add polynomials, combine like terms.
Add the polynomials.
Example 1
A
(4x 2 - x 3 + 2 + 5x 4) + (-x + 6x 2 + 3x 4) 5x4 -x 3
+4x 2
+2
Write in standard form.
+3x +6x -x ___ 4 3 2 8x -x +10x -x +2 4
B
2
Align like terms. Add.
(10x - 18x 3 + 6x 4 - 2) + (-7x 4 + 5 + x + 2x 3) (6x 4 - 18x 3 + 10x - 2) + (-7x 4 + 2x 3 + x + 5)
(
= 6x 4 =
) (
) +(
- 16x 3 +
+ 2x 3 +
Write in standard form.
) (
+ x + -2 +
+3
)
Group like terms. Add.
Reflect
2.
Is the sum of two polynomials always a polynomial? Explain.
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Your Turn
Add the polynomials. 3.
(17x 4 + 8x 2 - 9x 7 + 4 - 2x 3 ) + (11x 3 - 8x 2 + 12)
4.
(-8x + 3x 11 + x 6) + (4x 4 - x + 17)
Module 6
316
Lesson 1
Explain 2
Subtracting Polynomials
To subtract polynomials, combine like terms. Example 2
A
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.
B
12x
3
-8x 2
+5x
+19
+18x -6x +9x -3 ___ 30x 3 -14x 2 +14x +16 3
2
(-4x 2 + 8x 3 + 19 - 5x 5) - (9 + 2x 2 + 10x 5) Write in standard form and add the opposite.
(-5x 5 + 8x 3 - 4x 2 + 19) + (-10x 5 - 2x 2 - 9)
Group like terms
= -5x 5 -
Add
=
(
) +(
+ 8x 3 -
)+(
+ 10
) (
- 2x 2 +
)
-9
Reflect
5.
Is the difference of two polynomials always a polynomial? Explain.
Your Turn
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Subtract the polynomials. 6.
(23x 7 - 9x 4 + 1) - (-9x 4 + 6x 2 - 31)
7.
(7x 3 + 13x - 8x 5 + 20x 2) - (-2x 5 + 9x 2)
Module 6
317
Lesson 1
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
A
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.
(-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
= -24.2x 3 + 129.5x 2 + 137.6x + 16,200.2 The year 2009 is 4 years after 2005, so substitute 4 for x. ª)PVHIUPO.JGGMJO)BSDPVSU1VCMJTIJOH$PNQBOZt*NBHF$SFEJUTª+VUUB 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.
B
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.
Subtract the polynomials.
(41.5x 3 - 689.1x 2 + 4323.3x + 2796.6)
(-41.6x 3 + 560.9x 2 - 1632.7x + 6837.4)
= (41.5x 3 - 689.1x 2 + 4323.3x + 2796.6) + (41.6x 3 - 560.9x 2 + 1632.7x - 6837.4)
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Lesson 1
(
= 41.5x 3 + =
x3 -
)+(
x2 +
) (
- 560.9x 2 + x-
) (
)
+ 1632.7x + 2796.6 -
for x.
The year 2008 is 8 years after 2000, so substitute 3 2 83.1(8) - 1250(8) + 5956(8) - 4040.8 ≈
About
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.
Elaborate
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9.
How is the degree of a polynomial related to the degrees of the monomials that comprise the polynomial?
10. How is polynomial subtraction based on polynomial addition?
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?
12. Essential Question Check-In What is the result of adding or subtracting polynomials?
Module 6
319
Lesson 1
Evaluate: Homework and Practice 1.
Write the polynomial -23x 7 + x 9 - 6x 3 + 10 + 2x 2 in standard form, and then identify the degree and leading coefficient.
t0OMJOF)PNFXPSL t)JOUTBOE)FMQ t&YUSB1SBDUJDF
Add the polynomials.
(82x 8 + 21x 2 - 6) + (18x + 7x 8 - 42x 2 + 3)
3.
(15x - 121x 12 + x 9 - x 7 + 3x 2) + (x 7 - 68x 2 - x 9)
4.
(16 - x 2) + (-18x 2 + 7x 5 - 10x 4 + 5)
5.
(x + 1 - 3x 2) + (8x + 21x 2 - 1)
6.
(64 + x 3 - 8x 2) + (7x + 3 - x 2) + (19x 2 - 7x - 2)
7.
(x 4 - 7x 3 + 2 - x) + (2x 3 - 3) + (1 - 5x 3 - x 4 + x)
© Houghton Mifflin Harcourt Publishing Company
2.
Subtract the polynomials. 8.
(-2x + 23x 5 + 11) - (5 - 9x 3 + x)
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320
Lesson 1
9.
(7x 3 + 68x 4 - 14x + 1) - (-10x 3 + 8x + 23)
10. (57x 18 - x 2) - (6x - 71x 3 + 5x 2 + 2)
11. (9x - 12x 3) - (5x 3 + 7x - 2)
12. (3x 5 - 9) - (11 + 13x 2 - x 4) - (10x 2 + x 4)
13.
(10x 2 - x + 4) - (5x + 7) + (6x - 11)
Find the polynomial that models the problem and use it to estimate the quantity.
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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.
Module 6
321
Lesson 1
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.
16. 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?
Module 6
322
Lesson 1
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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.
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 amount the United States imported and exported in 2012.
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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.
Module 6
323
Lesson 1
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.
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
g. 4x 4x 2
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)
Module 6
324
© Houghton Mifflin Harcourt Publishing Company
= 15x - 17x 2y 2 + 16y + 7xy
Lesson 1
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?
24. 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.
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b. If the leg length is increased by 5 inches, will the perimeter also increase? By how much?
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.
Module 6
325
Lesson 1
Lesson Performance Task The table shows the average monthly maximum and minimum temperatures for Death Valley throughout one year.
Month
Maximum Temperature
Minimum Temperature
+BOVBSZ
67
40
February
73
46
March
82
55
April
91
62
May
101
73
+VOF
110
81
+VMZ
116
88
August
115
86
4FQUFNCFS
107
76
October
93
62
November
77
48
December
65
38
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?
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Module 6
326
Lesson 1
