Factoring Polynomials

1. Factoring Polynomials Algebra I

2. Vocabulary • Factors – The numbers used to find a product. • Prime Number – A whole number greater than one and its only factors are 1 and itself. • Composite Number – A whole number greater than one that has more than 2 factors.

3. Vocabulary • Factored Form – A polynomial expressed as the product of prime numbers and variables. • Prime Factoring – Finding the prime factors of a term. • Greatest Common Factor (GCF) – The product of common prime factors.

4. Prime or Composite? Ex) 36 Ex) 23

5. Prime or Composite? Ex) 36 Composite. Factors: 1,2,3,4,6,9,12,18,36 Ex) 23 Prime. Factors: 1,23

6. Prime Factorization Ex) 90 = 2 ∙ 45 = 2∙ 3∙ 15 = 2∙ 3 ∙ 3 ∙ 5 OR use a factor tree: 90 9 10 3 3 2 5

7. Prime Factorization of Negative Integers Ex) -140 = -1 ∙ 140 = -1 ∙ 2 ∙ 70 = -1 ∙ 2 ∙ 7 ∙ 10 = -1 ∙ 2 ∙ 7 ∙ 2 ∙ 5

8. Now you try… Ex) 96 Ex) -24

9. Now you try… Ex) 96 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 Ex) -24 -1 ∙ 2 ∙ 2 ∙ 2 ∙ 3

10. Prime Factorization of a Monomial 12a²b³= 2 · 2 · 3 · a · a · b · b · b -66pq²= -1 · 2 · 3 · 11 · p · q · q

11. Finding GCF Ex) 48 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 60 = 2 ∙ 2 ∙ 3 ∙ 5 GCF = 2 · 2 · 3 = 12 Ex) 15 = 3 · 5 16 = 2 · 2 · 2 · 2 GCF – none = 1

12. Now you try… Ex) 36x²y 54xy²z

13. Now you try… Ex) 36x²y = 2 · 2 · 3 · 3 · x · x · y 54xy²z = 2 · 3 · 3 · 3 · x · y · y · z GCF = 18xy

14. Factoring Using the (Reverse) Distributive Property • Factoring a polynomial means to find its completely factored form.

15. Factoring Using the (Reverse) Distributive Property • First step is to find the prime factors of each term. Ex) 12a²+ 16a 12a²= 2 · 2 · 3 · a · a 16a = 2 · 2 · 2 · 2 · a

16. Factoring Using the (Reverse) Distributive Property • First step is to find the prime factors of each term. • Next step is to find the GCF of the terms in the polynomial. Ex) 12a²+ 16a 12a²= 2 · 2 · 3 · a · a 16a = 2 · 2 · 2 · 2 · a GCF = 4a

17. Factoring Using the (Reverse) Distributive Property • First step is to find the prime factors of each term. • Next step is to find the GCF of the terms in the polynomial. • Now write what is left of each term and leave in parenthesis. Ex) 12a²+ 16a 12a²= 2 · 2 · 3 · a · a 16a = 2 · 2 · 2 · 2 · a 4a(3a + 4)

18. Factoring Using the (Reverse) Distributive Property • First step is to find the prime factors of each term. • Next step is to find the GCF of the terms in the polynomial. • Now write what is left of each term and leave in parenthesis. Ex) 12a²+ 16a 12a²= 2 · 2 · 3 · a · a 16a = 2 · 2 · 2 · 2 · a 4a(3a + 4) Final Answer 4a(3a + 4)

19. Another Example: 18cd²+ 12c²d + 9cd

20. Another Example: 18cd²+ 12c²d + 9cd 18cd² = 2 · 3 · 3 · c · d · d 12c²d = 2 · 2 · 3 · c · c · d 9cd = 3 · 3 · c · d GCF = 3cd Answer: 3cd(6d + 4c + 3)

21. FOIL Review Using FOIL: First Outer (x + 2)(x – 3) Inner Last x²+ -3x + 2x + -6 x²+ -1x + -6

22. Factoring by Grouping • Factor some polynomials having 4 or more terms. Pairs of terms are grouped together and factored using GCF. Ex) 4ab + 8b + 3a + 6

23. Factoring by Grouping • Factor some polynomials having 4 or more terms. Pairs of terms are grouped together and factored using GCF. Ex) 4ab + 8b + 3a + 6 (4ab + 8b) + (3a + 6) 4b(a + 2) + 3(a + 2) These must be the same! (a + 2)(4b + 3) *** check by using FOIL

24. Grouping more than one way • Group more than one way to get the same answer. • Use the commutative property to move terms to group. Ex) 4ab + 8b + 3a + 6

25. Grouping more than one way • Group more than one way to get the same answer. • Use the commutative property to move terms to group. Ex) 4ab + 8b + 3a + 6 4ab + 3a + 8b + 6 (4ab + 3a) + (8b + 6) a(4b + 3) + 2(4b + 3) (4b + 3)(a + 2)

26. Additive Inverse Property • Group with common factors. • Use inverse property to match up the factors. Ex) 35x – 5xy + 3y – 21

27. Additive Inverse Property • Group with common factors. • Use inverse property to match up the factors. Ex) 35x – 5xy + 3y – 21 (35x – 5xy) + (3y – 21) 5x(7 – y) + 3(y – 7) inverse property 5x(-1)(y – 7) + 3(y – 7) (7 – y) = -1(y – 7) -5x(y – 7) + 3(y – 7) (-5x + 3)(y – 7)

28. Factoring Trinomials When a=1 • ALWAYS check for GCF first! • Factor trinomials in the standard form ax²+ bx + c • Solve equations in the standard form ax²+ bx + c = 0

29. Factoring when b and c are positive x²+ 6x + 8 • factors(M) sum(A) 1, 8 9 2, 4 6 • 2 and 4 multiply to give you 8 and add together to give you 6. • Answer: (x+2)(x+4) • Check using FOIL

30. Factoring when b is negative and c is positive • Both factors need to be negative to have a positive product and a negative sum. x²- 10x + 16

31. Factoring when b is negative and c is positive • Both factors need to be negative to have a positive product and a negative sum. x²- 10x + 16 M A -1,-16 -17 -2,-8 -10 -4,-4 -8

32. Factoring when b is negative and c is positive • Both factors need to be negative to have a positive product and a negative sum. x²- 10x + 16 M A . -1,-16 -17 -2,-8 -10 -4,-4 -8 Answer: (x-2)(x-8)

33. Factoring when b is positive and c is negative • One factor has to be positive and one has to be negative to get a negative product.x²+ x – 12

34. Factoring when b is positive and c is negative • One factor has to be positive and one has to be negative to get a negative product.x²+ x – 12 M A 1,-12 -11 -1, 12 11 2, -6 -4 -2, 6 4 3,-4 -1 -3, 4 1

35. Factoring when b is positive and c is negative • One factor has to be positive and one has to be negative to get a negative product.x²+ x – 12 M A 1,-12 -11 -1, 12 11 2, -6 -4 -2, 6 4 3,-4 -1 -3, 4 1 Answer: (x-3)(x+4)

36. Factoring when b is negative and c is negative • One factor has to be positive and one has to be negative to get a negative product. x²-7x – 18

37. Factoring when b is negative and c is negative • One factor has to be positive and one has to be negative to get a negative product. x²-7x – 18 M A 1,-18 -17 -1, 18 17 2,-9 -7 -2, 9 7 3,-6 -3 -3, 6 3

38. Factoring when b is negative and c is negative • One factor has to be positive and one has to be negative to get a negative product. x²-7x – 18 M A 1,-18 -17 -1, 18 17 2,-9 -7 -2, 9 7 3,-6 -3 -3, 6 3 Answer: (x+2)(X-9)

39. Now you try… 3x² + 24x + 45

40. Now you try… 3x² + 24x + 45 3(x²+ 8x + 15) GCF 3(x + 3)(x + 5) final answer

41. Factoring Trinomials when a>1 • Multiply a and c. • Need to find two numbers where the product is equal to a∙c(30) and the sum is equal to b (17). 6x²+ 17x + 5

42. Factoring Trinomials when a>1 • Multiply a and c. • Need to find two numbers where the product is equal to a∙c(30) and the sum is equal to b (17). 6x²+ 17x + 5 M A 1, 30 31 2, 15 17 3, 10 13 5, 6 11

43. Factoring Trinomials when a>1 2, 15 product = 30, sum = 17 6x²+ 17x + 5 • Re write the first and last terms. 6x² + 5 • Fill in the middle with the two numbers you found, followed by the variable. 6x²+ 2x + 15x + 5 • Now factor by grouping.

44. Factoring Trinomials when a>1 (6x²+ 2x) + (15x + 5) group 2x(3x + 1) + 5(3x + 1) GCF (3x + 1)(2x + 5) final answer ***check by using FOIL

45. Now you try… 10x²- 43x + 28

46. Now you try… 10x²- 43x + 28 280-43 MA -2,-140 -142 -4,-70 -74 -8,-35 -43 -10,-28 -38 -14,-20 -34

47. Now you try… 10x²- 43x + 28 280-43 MA -2,-140 -142 -4,-70 -74 -8,-35 -43 -10,-28 -38 -14,-20 -34

48. Now you try… 10x²- 43x + 28 (10x²-8x) + (-35x + 28)

49. Now you try… 10x²- 43x + 28 (10x²-8x) + (-35x + 28) 2x(5x – 4) + 7(-5x + 4)

50. Now you try… 10x²- 43x + 28 (10x²-8x) + (-35x + 28) 2x(5x – 4) + 7(-5x + 4) 2x(5x – 4) + (-1)(7(5x – 4))