Just as you can perform operations on numbers, you can perform operations on polynomials. To add or subtract polynomials, combine like terms.

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Just as you can perform operations on numbers,
you can perform operations on polynomials. To add
or subtract polynomials, combine like terms.
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Additional Example 1 Adding and Subtracting
Monomials
Add or subtract.
A. 12p3 11p2 8p3
Identify like terms.
12p3 11p2 8p3
Rearrange terms so that like terms are together.
12p3 8p3 11p2
20p3 11p2
Combine like terms.
B. 5x2 6 3x 8
Identify like terms.
5x2 6 3x 8
Rearrange terms so that like terms are together.
5x2 3x 8 6
5x2 3x 2
Combine like terms.
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Additional Example 1 Adding and Subtracting
Monomials
Add or subtract.
C. t2 2s2 4t2 s2
Identify like terms.
t2 2s2 4t2 s2
Rearrange terms so that like terms are together.
t2 4t2 2s2 s2
3t2 s2
Combine like terms.
D. 10m2n 4m2n 8m2n
10m2n 4m2n 8m2n
Identify like terms.
6m2n
Combine like terms.
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Check It Out! Example 1
Add or subtract.
a. 2s2 3s2 s
2s2 3s2 s
Identify like terms.
5s2 s
Combine like terms.
b. 4z4 8 16z4 2
Identify like terms.
4z4 8 16z4 2
Rearrange terms so that like terms are together.
4z4 16z4 8 2
20z4 6
Combine like terms.
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Check It Out! Example 1
Add or subtract.
c. 2x8 7y8 x8 y8
Identify like terms.
2x8 7y8 x8 y8
Rearrange terms so that like terms are together.
2x8 x8 7y8 y8
x8 6y8
Combine like terms.
d. 9b3c2 5b3c2 13b3c2
Identify like terms.
9b3c2 5b3c2 13b3c2
b3c2
Combine like terms.
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Check up p. 441 s 3,4,6
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Polynomials can be added in either vertical or
horizontal form.
In vertical form, align the like terms and add
7x2 9x 3
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In horizontal form, use the Associative and
Commutative Properties to regroup and combine
like terms.
(5x2 4x 1) (2x2 5x 2)
(5x2 2x2) (4x 5x) (1 2)
7x2 9x 3
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Additional Example 2 Adding Polynomials
Add.
A. (4m2 5) (m2 m 6)
(4m2 5) (m2 m 6)
Identify like terms.
Group like terms together.
(4m2 m2) (m) (5 6)
5m2 m 11
Combine like terms.
B. (10xy x) (3xy y)
Identify like terms.
(10xy x) (3xy y)
Group like terms together.
(10xy 3xy) x y
7xy x y
Combine like terms.
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Additional Example 2 Adding Polynomials
Add.
C. (6x2 4y) (3x2 3y 8x2 2y)
(6x2 4y) (3x2 3y 8x2 2y)
Identify like terms.
Group like terms together within each polynomial.
(6x2 4y) (3x2 8x2 3y 2y)
(6x2 4y) (5x2 y)
Combine like terms in the second polynomial.
Use the vertical method.
x2 3y
Combine like terms.
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Additional Example 2 Adding Polynomials
Add.
D.
Identify like terms.
Group like terms together.
Combine like terms.
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Check It Out! Example 2
Add (5a3 3a2 6a 12a2) (7a3 10a).
(5a3 3a2 6a 12a2) (7a3 10a)
Identify like terms.
Group like terms together.
(5a3 7a3) (3a2 12a2) (10a 6a)
12a3 15a2 16a
Combine like terms.
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Check up p. 441 s 9, 10
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To subtract polynomials, remember that
subtracting is the same as adding the opposite.
To find the opposite of a polynomial, you must
write the opposite of each term in the polynomial
(2x3 3x 7) 2x3 3x 7
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Additional Example 3A Subtracting Polynomials
Subtract.
(x3 4y) (2x3)
Rewrite subtraction as addition of the opposite.
(x3 4y) (2x3)
(x3 4y) (2x3)
Identify like terms.
(x3 2x3) 4y
Group like terms together.
x3 4y
Combine like terms.
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Additional Example 3B Subtracting Polynomials
Subtract.
(7m4 2m2) (5m4 5m2 8)
Rewrite subtraction as addition of the opposite.
(7m4 2m2) (5m4 5m2 8)
(7m4 2m2) (5m4 5m2 8)
Identify like terms.
Group like terms together.
(7m4 5m4) (2m2 5m2) 8
2m4 3m2 8
Combine like terms.
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Additional Example 3C Subtracting Polynomials
Subtract.
(10x2 3x 7) (x2 9)
(10x2 3x 7) (x2 9)
Rewrite subtraction as addition of the opposite.
(10x2 3x 7) (x2 9)
Identify like terms.
Use the vertical method.
Write 0x as a placeholder.
11x2 3x 16
Combine like terms.
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Additional Example 3D Subtracting Polynomials
Subtract.
(9q2 3q) (q2 5)
Rewrite subtraction as addition of the opposite.
(9q2 3q) (q2 5)
(9q2 3q) (q2 5)
Identify like terms.
Use the vertical method.
Write 0 and 0q as placeholders.
8q2 3q 5
Combine like terms.
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Check It Out! Example 3
Subtract.
(2x2 3x2 1) (x2 x 1)
Rewrite subtraction as addition of the opposite.
(2x2 3x2 1) (x2 x 1)
(2x2 3x2 1) (x2 x 1)
Identify like terms.
Use the vertical method.
Write 0x as a placeholder.
2x2 x
Combine like terms.
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Check up p. 441 s 12, 14
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Additional Example 4 Application
A farmer must add the areas of two plots of land
to determine the amount of seed to plant. The
area of plot A can be represented by 3x2 7x
5, and the area of plot B can be represented by
5x2 4x 11. Write a polynomial that represents
the total area of both plots of land.
(3x2 7x 5)
Plot A.
(5x2 4x 11)
Plot B.

Combine like terms.
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Check It Out! Example 4
The profits of two different manufacturing plants
can be modeled as shown, where x is the number of
units produced at each plant.
Use the information above to write a polynomial
that represents the total profits from both
plants.
0.03x2 25x 1500
Eastern plant profits

0.02x2 21x 1700
Southern plant profits
Combine like terms.
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Check up p. 441 s 15