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Simplify each expression by combining like terms.

1.4x + 2x

2. 3y + 7y

3. 8p – 5p

4. 5n + 6n2

Simplify each expression.

5. 3(x + 4)

6. –2(t + 3)

7. –1(x2 – 4x – 6)

6x

10y

3p

not like terms

3x + 12

–2t – 6

–x2 + 4x + 6

Add and subtract polynomials.

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

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.

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.

Like terms are constants or terms with the same variable(s) raised to the same power(s). To review combining like terms, see lesson 1-7.

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.

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.

+ 2x2+ 5x+ 2

7x2+9x+3

Polynomials can be added in either vertical or horizontal form.

In vertical form, align the like terms and add:

In horizontal form, use the Associative and Commutative Properties to regroup and combine like terms.

(5x2 + 4x + 1) + (2x2 + 5x+ 2)

= (5x2 + 2x2 + 1) + (4x + 5x) + (1 + 2)

= 7x2+ 9x+ 3

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.

+ –5x2+ y

Example 2C: Adding Polynomials

Add.

(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)

(6x2– 4y) + (3x2+ 3y –8x2– 2y)

Identify like terms.

Group like terms together within each polynomial.

(6x2 + 3x2 – 8x2) + (3y – 4y – 2y)

Use the vertical method.

Combine like terms.

x2– 3y

Simplify.

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.

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

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.

Example 3B: Subtracting Polynomials

Subtract.

(7m4 – 2m2) – (5m4 – 5m2 + 8)

(7m4 – 2m2) + (–5m4+5m2– 8)

Rewrite subtraction as addition of the opposite.

(7m4– 2m2) + (–5m4+ 5m2 – 8)

Identify like terms.

Group like terms together.

(7m4– 5m4) + (–2m2+ 5m2) – 8

2m4 + 3m2 – 8

Combine like terms.

–x2 + 0x+ 9

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.

+− q2– 0q + 5

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.

+ –x2 – x – 1

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.

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.

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 profit.

+

–0.02x2 + 21x – 1700

Southern plant profit.

Combine like terms.

Add or subtract.

1. 7m2 + 3m + 4m2

2. (r2 + s2) – (5r2 + 4s2)

3. (10pq + 3p) + (2pq – 5p + 6pq)

4. (14d2 – 8) + (6d2 – 2d +1)

11m2 + 3m

(–4r2 – 3s2)

18pq – 2p

20d2 – 2d – 7

5. (2.5ab + 14b) – (–1.5ab + 4b)

4ab + 10b

6. A painter must add the areas of two walls to determine the amount of paint needed. The area of the first wall is modeled by 4x2 + 12x + 9, and the area of the second wall is modeled by

36x2 – 12x + 1. Write a polynomial that represents the total area of the two walls.

40x2 + 10

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