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Adding and Subtracting Polynomials. 7-6. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra 1. Warm Up Simplify each expression by combining like terms. 1. 4 x + 2 x 2. 3 y + 7 y 3. 8 p – 5 p 4. 5 n + 6 n 2 Simplify each expression. 5. 3( x + 4) 6. –2( t + 3)

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## 7-6

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7-6

Warm Up

Lesson Presentation

Lesson Quiz

Holt Algebra 1

Warm Up

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

Objective

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

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

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.

Remember!

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.

Check It Out! Example 1

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.

Check It Out! Example 1

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.

5x2+ 4x+1

+ 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

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.

6x2– 4y

+ –5x2+ y

(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.

Identify like terms.

Group like terms together.

Combine like terms.

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.

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.

–10x2 – 3x + 7

–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.

9q2 – 3q+ 0

+− 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+ 0x + 1

+ –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.

8x2 + 3x + 6

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.

–0.05x2 + 46x – 3200

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.

Lesson Quiz: Part I

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

Lesson Quiz: Part II

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