1. Warm Up
Combine like terms.
1. 9x + 4x 2. –3y + 7y
3. 7n + (–8n) + 12n
Find the perimeter of each rectangle.
4. a 10 ft by 12 ft rectangle
5. a 5 m by 8 m rectangle
Simplify.
6. 3(2x2 – x) + x2+ 1
13x 4y
11n
44 ft
26 m
7x2 – 3x + 1

2. Learn to add polynomials.

3. Example 1A: Adding Polynomials Horizontally
(5x3 + x2 + 2) + (4x3 + 6x2)
(5x + x + 2) + (4x + 6x ) 3 2
5x + x x + 6x 3 2
Associative Property
9x + 7x + 2 3 2
Combine like terms.

4. Example 1B: Adding Polynomials Horizontally
(6x3+ 8y2 + 5xy) + (4xy – 2y2)
(6x3+ 8y2+ 5xy) + (4xy – 2y2)
6x3 + 8y2 + 5xy + 4xy – 2y2
Associative Property
6x + 6y + 9xy 3 2
Combine like terms.

5. Example 1C: Adding Polynomials Horizontally
(3x2y – 5x) + (4x + 7) + 6x2y
(3x2y – 5x) + (4x + 7) + 6x2y
3x2y – 5x + 4x x2y
Associative Property
9x2y – x + 7
Combine like terms.

6. Example 2A Add. (3y4 + y2 + 6) + (5y4 + 2y2) (3y + y + 6) + (5y + 2y )
Associative Property
8y + 3y + 6 4 2
Combine like terms.

7. Example 2B Add. (9x3 + 6p2 + 3xy) + (8xy – 3p2)
Associative Property
9x3+ 3p2 + 11xy
Combine like terms.

8. Example 2C
Add.
(3z2w – 5x) + (2x + 8) + 6z2w
(3z2w – 5x) + (2x + 8) + 6z2w
3z2w – 5x + 2x z2w
Associative Property
9z2w – 3x + 8
Combine like terms.

9. You can also add polynomials in a vertical format
You can also add polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms. If the terms are rearranged, remember to keep the correct sign with each term.

10. Example 3: Adding Polynomials Vertically
A. (4x2 + 2x + 11) + (2x2 + 6x + 9)
4x2 + 2x + 11
+ 2x2 + 6x + 9
Place like terms in columns.
6x2 + 8x + 20
Combine like terms.

11. Example 3: Adding Polynomials Vertically
B. (3mn2 – 6m + 6n) + (5mn2 + 2m – n)
C. (–x2y2 + 5x2) + (–2y2 + 2) + (x2 + 8)
3mn2 – 6m + 6n
+ 5mn2 + 2m – n
Place like terms in columns.
8mn2 – 4m + 5n
Combine like terms.
–x2y2 + 5x2
–2y2 + 2
Place like terms in columns. x
–x2y2 + 6x2 – 2y2 + 10
Combine like terms.

12. Example 4
Add.
A. (6x2 + 6x + 13) + (3x2+ 2x + 4)
6x2 + 6x + 13
+ 3x2 + 2x + 4
Place like terms in columns.
9x2 + 8x + 17
Combine like terms.

13. Example 4
Add.
B. (4mn2 + 6m + 2n) + (2mn2 – 2m – 2n)
C. (x2y2 – 5x2) + (2y2 – 2) + (x2)
4mn2 + 6m + 2n
+ 2mn2 – 2m – 2n
Place like terms in columns.
6mn2 + 4m
Combine like terms.
x2y2 – 5x2
2y2 – 2
Place like terms in columns. x2
x2y2 – 4x2 + 2y2 – 2
Combine like terms.

14. Subtraction is the opposite of addition. To
subtract a polynomial, you need to find its
opposite.

15. Example 1: Finding the Opposite of a Polynomial
Find the opposite of each polynomial.
A. 8x3y4z2
–(8x3y4z2)
–8x3y4z2
Distributive Property.
B. –3x4 + 8x2
–(–3x4 + 8x2)
3x4 – 8x2
Distributive Property.

16. Additional Example 1: Finding the Opposite of a Polynomial
Find the opposite of the polynomial.
C. 9a6b4 + a4b2 – 1
–(9a6b4 + a4b2 – 1)
–9a6b4 – a4b2 + 1
Distributive Property.

17. To subtract a polynomial, add its opposite.

18. Example 1: Subtracting Polynomials Horizontally
A. (5x2 + 2x – 3) – (3x2 + 8x – 4)
Add the
opposite.
= (5x2 + 2x – 3) + (–3x2 – 8x + 4)
Associative
property.
= 5x2 + 2x – 3 – 3x2 – 8x + 4
= 2x2 – 6x + 1
Combine like
terms.

19. Example 1: Subtracting Polynomials Horizontally
B. (b2 + 4b – 1) – (7b2 – b – 1)
= (b2 + 4b – 1) + (–7b2 + b + 1)
Add the opposite.
Associative
property.
= b2 + 4b – 1 – 7b2 + b + 1
= –6b2 + 5b
Combine like
terms.

20. Example 2A
Subtract.
(2y3 + 3y + 5) – (4y3 + 3y + 5)
Add the
opposite.
= (2y3 + 3y + 5) + (–4y3 – 3y – 5)
Associative
property.
= 2y3 + 3y + 5 – 4y3 – 3y – 5
= –2y3
Combine like
terms.

21. Example 2B
Subtract.
(c3 + 2c2 + 3) – (4c3 – c2 – 1)
= (c3 + 2c2 + 3) + (–4c3 + c2 + 1)
Add the opposite.
= c3 + 2c2 + 3 – 4c3 + c2 + 1
Associative
property.
= –3c3 + 3c2 + 4
Combine like
terms.

22. You can also subtract polynomials in a vertical format
You can also subtract polynomials in a vertical format. Write the second polynomial below the first one, lining up the like terms.

23. Example 3: Subtracting Polynomials Vertically
(2n2 – 4n + 9) – (6n2 – 7n + 5)
(2n2 – 4n + 9)
2n2 – 4n + 9
– (6n2 – 7n + 5)
+ –6n2 + 7n – 5
Add the
opposite.
–4n2 + 3n + 4

24. Example 4: Subtracting Polynomials Vertically
(10x2 + 2x – 7) – (x2 + 5x + 1)
(10x2 + 2x – 7)
10x2 + 2x – 7
– (x2 + 5x + 1)
+ –x2 – 5x – 1
Add the
opposite.
9x2 – 3x – 8

25. Example 5: Subtracting Polynomials Vertically
(6a4 – 3a2 – 8) – (–2a4 + 7)
(6a4 – 3a2 – 8)
6a4 – 3a2 – 8
– (–2a4 + 7)
+ 2a – 7
Rearrange as needed.
8a4 – 3a2 – 15

26. Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems

27. Lesson Quiz: Part I Add. 1. (2m2 – 3m + 7) + (7m2 – 1)
2. (yz2 + 5yz + 7) + (2yz2 – yz) 3.
9m2 – 3m + 6
3yz2+ 4yz + 7
(2xy2 + 2x – 6)
+ (5xy2 + 3y + 8)
7xy + 2x + 3y + 2 2

28. Lesson Quiz
Find the opposite of each polynomial.
Subtract.
3. (3z2 – 7z + 6) – (2z2 + z – 12)
1. 3a2b2c3
–3a2b2c3
2. –3m3 + 2m2n
3m3 – 2m2n
z2 – 8z + 18
4. –18h3 – (4h3 + h2 – 12h + 2)
5. (3b2c + 5bc2 – 8b2)
– (4b2c + 2bc2 – c2)
– 22h3 – h2 + 12h – 2
– b2c + 3bc2 – 8b2 + c2

29. Lesson Quiz for Student Response Systems
1. Add (4p2 – 8p +11) + (6p2 – 9).
A. 10p2 – 8p + 2
B. 10p2 + 8p + 20
C. 2p2 + 8p + 2
D. 10p2 – 8p + 20

30. Lesson Quiz for Student Response Systems
2. Add (gh2 + 9gh + 11) + (3gh2 – gh).
A. 2gh2 + 8gh + 11
B. 2gh2 + 10gh + 11
C. 4gh2 + 8gh + 11
D. 4gh2 + 10gh + 11

31. Lesson Quiz for Student Response Systems
3. Add (7uv3 + 11u) + (6uv3 – u – 9) + (4u – 2).
A. 13uv3 – 6u – 7
B. uv2 – 16u + 7
C. uv3 + 14u – 11
D. 13uv3 + 14u – 11

32. Lesson Quiz for Student Response Systems
3. Subtract.
(11p2 – 5p + 9) – (3p2 + p – 17)
A. 14p2 + 4p – 8
B. 14p2 – 6p + 26
C. 8p2 – 6p + 26
D. 8p2 + 4p – 8