1. Adding and Subtracting Polynomials 7-7
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

2. 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

3. Objective
Add and subtract polynomials.

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

5. Example 1: Adding and Subtracting Monomials
Add or subtract.
A. 12p3 + 11p2 + 8p3
12p3 + 11p2 + 8p3
Identify like terms.
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.

6. Example 1: Adding and Subtracting Monomials
Add or subtract.
C. t2 + 2s2 – 4t2 – s2
t2 + 2s2 – 4t2 – s2
Identify like terms.
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.

7. 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.
Remember!

8. 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 – z4 + 2
4z4 – z4 + 2
Identify like terms.
Rearrange terms so that like terms are together.
4z4 + 16z4 – 8 + 2
20z4 – 6
Combine like terms.

9. 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
9b3c2 + 5b3c2 – 13b3c2
Identify like terms.
b3c2
Combine like terms.

10. 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
7x2 + 9x + 3
(5x2 + 4x + 1) + (2x2 + 5x + 2)
= (5x2 + 2x2) + (4x + 5x) + (1 + 2)
= 7x2 + 9x + 3

11. Example 2: Adding Polynomials
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)
(10xy + x) + (–3xy + y)
Identify like terms.
Group like terms together.
(10xy – 3xy) + x + y
7xy + x + y
Combine like terms.

12. Example 2C: Adding Polynomials
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)
(6x2 – 4y) + (3x2 + 3y – 8x2 – 2y)
Identify like terms.
Combine like terms in the second polynomial.
(6x2 – 4y) + (–5x2 + y)
(6x2 –5x2) + (–4y + y)
Combine like terms.
x2 – 3y
Simplify.

13. Example 2D: Adding Polynomials
Identify like terms.
Group like terms together.
Combine like terms.

14. Check It Out! Example 2
Add (5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a).
(5a3 + 3a2 – 6a + 12a2) + (7a3 – 10a)
Identify like terms.
(5a3 + 7a3) + (3a2 + 12a2) + (–10a – 6a)
Group like terms together.
12a3 + 15a2 – 16a
Combine like terms.

15. 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

16. Example 3A: Subtracting Polynomials
(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.

17. Example 3B: Subtracting Polynomials
(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.

18. Example 3C: Subtracting Polynomials
(–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.
–10x2 – 3x + 7
–x2 + 0x + 9
Use the vertical method.
Write 0x as a placeholder.
–11x2 – 3x + 16
Combine like terms.

19. Example 3D: Subtracting Polynomials
(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.
9q2 – 3q + 0
+ − q2 – 0q + 5
Write 0 and 0q as placeholders.
8q2 – 3q + 5
Combine like terms.

20. 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.
–x2 + 0x + 1
+ –x2 – x – 1
Write 0x as a placeholder.
–2x2 – x
Combine like terms.

21. 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 Write a polynomial that represents the total area of both plots of land.
(3x2 + 7x – 5)
Plot A. +
(5x2 – 4x + 11)
Plot B.
8x2 + 3x + 6
Combine like terms.

22. 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.
–0.05x2 + 46x – 3200
Combine like terms.

23. Lesson Quiz: Part I
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

24. 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