1. Chapter 10

2. Adding and Subtracting Polynomials
Chapter 10.1

3. Vocabulary Polynomial Standard form Degree Degree of a polynomial
Expression whose terms are of the form where k is a nonnegative integer.
Standard form
Degree
Exponent of the variable for each term
Degree of a polynomial
The largest degree of its terms
Leading coefficient
The coefficient of the first term

4. Classifying Polynomials
Degree
Classified by degree
Classified by number of terms 6
Constant
Monomial
-2x 1
Linear
3x + 1
Binomial
-x² + 2x – 5 2
Quadratic
Trinomial
4x³ - 8x 3
Cubic
2x - 7x³ - 5x + 1 4
Quartic 4

5. Adding Polynomials 1) 2)

6. Subtracting Polynomials1) 2) 3)

7. Adding and Subtracting Polynomials

9. Multiplying Polynomials
Chapter 10.2
Multiplying Polynomials

10. Multiply the Polynomials
Use the distributive property 1) 2) 4) 3)

11. Multiply the Polynomials
Use the distributive property 5) 6)

12. (x + 2)(x – 3) x 2 2x x x²
-3x -6 -3
x² - x – 6

13. (3x + 4)(x + 5) 3x 4 4x x
3x²
15x 20 5
3x² + 19x +20

14. (3x + 4)(2x + 1) 3x 4 8x 2x
6x² 3x 4 1
6x² + 11x +4

15. (3x + 10)(2x + 6) 3x 10
20x 2x
6x²
18x 60 6
6x² + 38x +60

16. 2x -5 4x² 8x³ -20x² -3x -6x² 15x -1 -2x 5 8x³ - 26x² + 13x + 5

17. x -2 5 5x -10 3x 3x² -6x -x² -x³ 2x² -x³ + 5x² - x – 10

19. Special Products of Polynomials
Chapter 10.3

20. (x + 3)²
= (x + 3)(x + 3) x 3 3x x x² 3x 9 3
x² + 6x + 9

21. (3x + 4)²
= (3x + 4)(3x + 4) 3x 4
12x 3x
9x²
12x 16 4
9x² + 24x + 16

22. (x – 2)²
= (x – 2)(x – 2) x -2
-2x x x²
-2x 4 -2
x² - 4x + 4

23. 2x -7y -14xy 2x 4x² 49y² -7y 4x² - 28x + 49y² -14xy

25. Dividing Polynomials
Chapter 11.7

26. 1) Divide 12x² - 20x + 8 by 4x.
2) Divide 8x + 14 by 2.
3) Divide 9c² + 3c by c.

27. 4) Divide -2x² - 12x by -2x.
5) Divide 9a² -54a – 36 by 3a.

29. Factoring x² + bx + c
Chapter 10.5

30. Factoring To factor x² + bx + c you need to find
numbers p and q such that
p + q = b and pq = c
x² + bx + c = (x + p)(x + q) when p + q = b and pq = c
Example:
x² + 6x + 8 = (x + 4)(x + 2) = 6 and 4(2) = 8

31. Factor x² + 3x + 2
Find the factors of 2 1 2 -1 -2
(x + 1)(x + 2)

32. Factor x² - 5x + 6 (x – 2)(x – 3) Find the factors of 6 1 6 -6 -1 2 3-2 -3
(x – 2)(x – 3)

33. Factor x² - 2x – 8 (x + 2)(x – 4) Find the factors of -8 1 -8 -1 8 2-4 -2 4
(x + 2)(x – 4)

34. Factor x² + 7x – 18 (x – 2)(x + 9) Find the factors of -18 1 -18 -1 18-9 -2 9 3 -6
(x – 2)(x + 9) -3 6

36. Factoring ax² + bx + c
Chapter 10.6

37. 2x² 5 x 5 2x 10x 1 x (x + 5)(2x + 1) Factor 2x² + 11x + 5 2(5) = 10 1-1
-10 1 x 5 2 -2 -5
(x + 5)(2x + 1)

38. 3x² - 7 3x -7 x -7x 1 3x (3x – 7)(x + 1) Factor 3x² - 4x – 7 3(-7)
= -21 x
-7x 1
-21 -1 21 1 3x 3 -7 -3 7
(3x – 7)(x + 1)

39. 6x² 15 3x -5 2x -10x -3 -9x (3x – 5)(2x – 3) Factor 6x² - 19x + 15
6(15)
= 90 3x -5 90 1
-90 2x -1
-10x 45 2
-45 -2 -3
-9x 30 3 6 15
-30 -3
-15 -6 5 18 9 10
(3x – 5)(2x – 3) -5
-18 -9
-10

40. 2(3x² -x – 4) 3x² - 4 3x -4 x -4x 1 3x 2(3x – 4)(x + 1)
Factor 6x² - 2x – 8
2(3x² -x – 4)
3x²
- 4 3x -4
3(-4)
= -12 x
-4x 1
-12 -1 12 1 2 -6 3x -2 6 3 -4 4 -3
2(3x – 4)(x + 1)

42. Factoring Special Products
Chapter 10.7

43. Factoring Special Products
Difference of Two Squares
a² - b² = (a + b)(a – b)
9x² - 16 = (3x + 4)(3x – 4)
Perfect Square Trinomial
a² + 2ab + b² = (a + b) ²
x² + 8x + 16 = (x + 4) ²
a² - 2ab + b² = (a – b) ²
x² - 12x + 36 = (x – 6) ²

44. Difference of Two Squares
m² - 4
(m + 2)(m – 2)
4p² - 25
(2p + 5)(2p – 5)
50 – 98x²
2(25 – 49x²)
2(5 + 7x)(5 – 7x)

45. Perfect Square Trinomial
x² - 4x + 4
(x – 2)²
16y² + 24y + 9
(4y + 3)²
3x² - 30x + 75
3(x² - 10x + 25)
3(x – 5)²

47. Factoring Using the Distributive Property
Chapter 10.8

48. Factoring Completely Find the GCF Factor out the GCF
Factor the remaining terms

49. Practice 3) 1) 2) 4) 5)

50. Factor by Grouping Group terms Factor each group
Use distributive property

51. Practice 2) 1) 3) 4)