1. Polynomial operations
Integrated mathematics

2. Types of Polynomials Name # Terms Example Monomial Binomial Trinomial

3. Like Terms
Same Variable
Same Exponent
Example:
2𝑥 2 and 3 𝑥 2

4. Degree: Largest exponent on polynomial
5 𝑥 3
3 𝑥 5 +4 𝑥 2
5 𝑥 3 +2 𝑥 2 +4

5. Descending Order: Highest degree first
3 𝑥 2 +4 𝑥 5 −7𝑥
5𝑦 −9−2 𝑦 4 −6 𝑦 3

6. Descending Order: 3 𝑥 4 𝑦 2 +2 𝑥 5 𝑦 2 −8 𝑥 3 𝑦 2
3 𝑥 4 𝑦 2 +2 𝑥 5 𝑦 2 −8 𝑥 3 𝑦 2
3 𝑥 3 𝑦−8𝑥 𝑦 3 +5 𝑥 4 𝑦 4

7. Try on your own 𝟐 𝒙 𝟑 −𝟓𝒙+𝟕 𝒙 𝟓 𝟒 𝒙 𝟒 𝒚−𝒙+𝟕 𝒙 𝟔 𝟐 𝒙 𝟑 −𝟓𝒙 𝒚 𝟒 +𝟕 𝒙 𝟕
4 𝒂 𝟔 +𝟗𝒙𝒚𝒛+𝟕 𝒙 𝟓

8. Example 1
3𝑥 2 +2𝑥−2 +( −2𝑥 2 +5𝑥+5)

9. Example 2
31𝑚 4 + 𝑚 2 +2𝑚−1 +( −7𝑚 4 + 5𝑚 2 −2𝑚+2)

10. Example 3
4𝑎 2 𝑏−5𝑎+2 +( −2𝑎 2 𝑏−2𝑎−4)

11. Example 4
3𝑛 3 − 3𝑚 3 𝑛 2 −5𝑛−3 +( 5𝑛 3 + 2𝑚 3 𝑛 2 −3𝑚−2𝑛−2)

12. Example 5
−2𝑚 3 − 5𝑚 2 −2𝑚−4 +( 𝑚 4 − 6𝑚 2 +7𝑚−10)

13. Example 6
−2𝑥 4 𝑦 3 −5𝑥𝑦+2 +( 𝑥 4 𝑦 3 + 𝑥 2 +2𝑥𝑦+5)

14. Try on your own 2𝑥 2 +7𝑥+3 +( −5𝑥 2 −3𝑥−6) 𝟒 𝒙 𝟒 −𝒙+𝟕 +(𝟗 𝒙 𝟒 +𝟑𝒙−𝟕)
𝟔 𝒙 𝟑 −𝟑𝒙 𝒚 𝟒 +𝟕 +(𝟐 𝒙 𝟑 −𝟓𝒙 𝒚 𝟒 +𝟕 𝒙 𝟕 )
𝟑 𝒙 𝟐 𝒛− 𝟓𝒚 𝟒 𝒙+𝟑𝒙 + −𝟐 𝒙 𝟐 𝒛− 𝒚 𝟒 𝒙−𝟐𝒙

15. PDN
Identify the mistake and correct the problem.
4𝑥 4 +3 𝑥 2 −3𝑥 + 2 𝑥 2 −5𝑥+2
=6 𝑥 2 −2 𝑥 2 −1𝑥

16. Subtract
Example 7
𝑎 3 −2 𝑎 2 +4 −( 𝑎 4 −4 𝑎 3 − 3𝑎 2 )

17. Example 8
3𝑥 3 𝑦 2 −4𝑥𝑦+1 −(−4 𝑥 3 𝑦 2 −3 𝑥 2 𝑦 2 +3𝑥𝑦−5)

18. Example 9
4 𝑥 3 +2 𝑥 2 −2𝑥−3 −(2 𝑥 3 −3 𝑥 2 +2)

19. Ticket out the door 2𝑥 2 +7𝑥+3 −( −5𝑥 2 −3𝑥−6)
𝟑 𝒙 𝟐 𝒛− 𝟓𝒚 𝟒 𝒙+𝟑𝒙 − −𝟐 𝒙 𝟐 𝒛− 𝒚 𝟒 𝒙−𝟐𝒙

20. PDN 2 𝑥 2 +2𝑥−7 − 2 𝑥 2 −2𝑥−3 2 𝑥 2 +2𝑥−7 + 2 𝑥 2 +2𝑥+3 =4 𝑥 2 +4𝑥−4
Find and describe the mistake. Then, correct.
2 𝑥 2 +2𝑥−7 − 2 𝑥 2 −2𝑥−3
2 𝑥 2 +2𝑥−7 + 2 𝑥 2 +2𝑥+3
=4 𝑥 2 +4𝑥−4

21. Multiplying Polynomials
Simplify using the distributive property
Example 1
2x ( 5x )

22. Simplify using the distributive property
Example 2
− 𝟓𝒙 𝟐 (𝟑𝒙+𝟓)

23. Simplify using the distributive property
Example 3
𝟑𝒂 𝟐 ( −𝟓𝒂 𝟑 +𝟐𝒂−𝟕)

24. Simplify using the distributive property
Example 4
8p( 𝟑𝒒 𝟒 −𝟐 𝒒 𝟑 𝒑 𝟐 +𝟐𝒑)

25. Try on your own
𝟑 𝒙−𝟕
𝟐𝒙 𝟐 ( 𝟓𝒙 𝟐 −𝟕𝒙+𝟏)
−𝟖𝒚(−𝟐𝒚 𝟐 −𝟕)

26. Ticket out the door
𝟓𝒕(𝟐𝒕−𝟒)
𝟒𝒙𝒚 𝟐 (𝟐 𝒙 𝟓 −𝟒𝒚)

27. PDN 3 𝑥 2 𝑦 3 𝑥 4 𝑦−4𝑥 =9 𝑥 2+4 𝑦−12 𝑥 2+1 𝑦 =9 𝑥 6 𝑦−12 𝑥 3 𝑦
Find and describe the mistake. Then, correct.
3 𝑥 2 𝑦 3 𝑥 4 𝑦−4𝑥
=9 𝑥 2+4 𝑦−12 𝑥 2+1 𝑦
=9 𝑥 6 𝑦−12 𝑥 3 𝑦

28. Multiplying Binomials
x ( x )
1 ( x )

29. How would you use the distributive property to simplify this?
Ex. 5
( x + 1) ( x )

30. F O I L
Ex. 6
( 2n + 3) ( n - 6 )

31. Ex. 7
(𝒄−𝟓)(𝟑𝒄+𝟏)

32. Ex. 8
(𝟐𝒙−𝟏)(𝟑𝒙−𝟏)

33. Ticket out the door
(𝟐𝒙−𝟓)(𝒙+𝟕)
(𝒃−𝟏)(𝒃−𝟗)

34. PDN (3𝑥−2) 𝑥−4 =3𝑥−12𝑥−2𝑥+8 =−11𝑥+8
Find and describe the mistake. Then, correct.
(3𝑥−2) 𝑥−4
=3𝑥−12𝑥−2𝑥+8
=−11𝑥+8

35. SPECIAL CASES 𝐚+𝐛 𝐚 −𝐛 = 𝐚 −𝐛 𝐚+𝐛 = 𝐚 𝟐 − 𝐛 𝟐
𝐚+𝐛 𝐚 −𝐛 = 𝐚 −𝐛 𝐚+𝐛 = 𝐚 𝟐 − 𝐛 𝟐
(𝒂+𝒃 ) 𝟐 = 𝒂 𝟐 + 2ab + 𝒃 𝟐
(𝒂 −𝒃) 𝟐 = 𝒂 𝟐 - 2ab + 𝒃 𝟐

36. Special Cases
Example 9
(𝒚+𝟓)(𝒚−𝟓)

37. Special Cases
Example 10
(𝟒𝒙+𝟕)(𝟒𝒙−𝟕)

38. Special Cases
Example 11
( 𝒚−𝟓) 𝟐

39. Special Cases
Example 12
( 𝟐𝒗 𝟐 −𝟖) 𝟐

40. Try on your own
(𝟐𝒙−𝟓) 𝟐
(𝒙−𝟓) 𝟐
(−𝟐𝒙+𝟕) 𝟐
(𝟑𝒙−𝟏) 𝟐

41. Binomials & Trinomials
Example 1
(𝒙+𝟏)( 𝒙 𝟐 +𝒙+𝟏)

42. Binomials & Trinomials
Example 2
( 𝒙 𝟐 +𝟒)( 𝒙 𝟐 +𝟐𝒙−𝟑)

43. Binomials & Trinomials
Example 3
(𝟑𝒙+𝟐)(𝟑 𝒙 𝟐 +𝟐𝒙−𝟏)

44. Binomials & Trinomials
Example 4
(𝟑𝒄−𝟒)( 𝟐𝒄 𝟐 −𝒄+𝟑)

45. Try on your own (𝟐𝒕−𝟖)( 𝟑𝒕 𝟐 −𝒕+𝟒) (𝟓𝒗+𝟐)( 𝟑𝒗 𝟐 +𝟐𝒗−𝟖)
( 𝟒𝒅 𝟐 +𝟑)( 𝟐𝒅 𝟐 +𝒅+𝟓)
(𝟑𝒃−𝟕)( 𝟓𝒃 𝟒 −𝟓𝒃−𝟗)