1. 10/16/2015Math 120 - KM1 Chapter 6: Introduction to Polynomials and Polynomial Functions 6.1 Introduction to Factoring 6.2 FactoringTrinomials: x 2 +bx+c 6.3 FactoringTrinomials: ax 2 +bx+c 6.4 Special Factoring 6.5 Factoring: A General Strategy 6.6 Applications

2. 10/16/2015Math 120 - KM2 6.1 Introduction to Factoring 6.1

3. 10/16/2015Math 120 - KM3 Let’s Build the Greatest Common Factor of 90x 2 y 3 z and 50y 4 z 5 The GCF of 90x 2 y 3 z and 50y 4 z 5 is the product of the “common” bases raised to the smallest exponent. or 6.1

4. 10/16/2015Math 120 - KM4 Let’s Build the Greatest Common Factor of 21x 2 z and 10y 4 21x 2 z and 10y 4 have no common factors! The only factor common to both expressions is 1. 21x 2 z and 10y 4 are RELATIVELY PRIME because their GCF is 1. 6.1

5. 10/16/2015Math 120 - KM5 Factoring out the GCF is Reversing the Distributive Property 6.1

6. 10/16/2015Math 120 - KM6 Factor out the GCF from 12x 5 + 20x 3 12x 5 20x 3 4x 3 3x 2 5 4x 3 3x 2 5 6.1

7. 10/16/2015Math 120 - KM7 Factor: 12x 5 + 20x 3 12x 5 + 20x 3 = 4x 3 (3x 2 + 5) 6.1

8. 10/16/2015Math 120 - KM8 Factor: 6x 2 y 5 - 8x 3 y 4 6x 2 y 5 - 8x 3 y 4 = 2x 2 y 4 (3y - 4x) 6.1

9. 10/16/2015Math 120 - KM9 Factor: 9x 3 – 11y 2 + 3 9x 3 – 11y 2 + 3 6.1

10. 10/16/2015Math 120 - KM10 Factoring 6.1

11. 10/16/2015Math 120 - KM11 Factor a Tricky One! x(x + 2) – 6(x + 2) = ( x + 2 )( x – 6 ) 6.1

12. 10/16/2015Math 120 - KM12 Another Tricky One! (x - 7)3x +(x - 7)5 = ( x - 7 )( 3x + 5 ) 6.1

13. 10/16/2015Math 120 - KM13 Factor by Grouping: Example 1: REVERSE FOIL ab + 7b – 3a – 21 = b(a + 7)– 3(a + 7) = (a + 7)(b - 3) (a + 7)(b – 3) = ab – 3a + 7b - 21 6.1

14. 10/16/2015Math 120 - KM14 Factor by Grouping: Example 2: REVERSE FOIL x 2 + 3x + 5x + 15 = x(x + 3) + 5(x + 3) = (x + 3)(x + 5) 6.1

15. 10/16/2015Math 120 - KM15 Factor by Grouping: Example 3: REVERSE FOIL x 2 + 5x – 5x - 25 = x(x + 5) - 5(x + 5) = (x + 5)(x - 5) 6.1

16. 10/16/2015Math 120 - KM16 Factor by Grouping: Example 4: REVERSE FOIL x 2 - 9x + 11x - 99 = x(x - 9) + 11(x - 9) = (x + 11)(x - 9) 6.1

17. 10/16/2015Math 120 - KM17 Factor by Grouping: Example 5: REVERSE FOIL x 3 - 10x 2 - 10x + 100 = x 2 (x - 10) - 10(x - 10) = (x 2 – 10)(x - 10) 6.1

18. 10/16/2015Math 120 - KM18 Factor by Grouping: Example 6: REVERSE FOIL 18x 2 - 21x + 30x - 35 = 3x(6x - 7) + 5(6x - 7) = (6x - 7)(3x + 5) 6.1

19. 10/16/2015Math 120 - KM19 Factor by Grouping: Example 7: REVERSE FOIL 25x 2 + 35x + 35x + 49 = 5x(5x + 7)+ 5(5x + 7) = (5x + 7) (5x + 7)

20. 10/16/2015Math 120 - KM20 Where We Left Off Last Class

21. 10/16/2015Math 120 - KM21 4.4 & 4.5 FactoringTrinomials: ax 2 +bx+c 6.2

22. 10/16/2015Math 120 - KM22 First, Let’s Review Factor by Grouping ab + 7b – 3a – 21 x 2 + 2x + 10x + 20 6.2

23. 10/16/2015Math 120 - KM23 Now, Let’s Review FOIL! Aha! FL = OI (8)(-15) = (10)(-12) -120 = - 120 6.2

24. 10/16/2015Math 120 - KM24 What’s the Diamond? ax 2 + bx + c Add to b Multiply to ac 6.2

25. 10/16/2015Math 120 - KM25 2x 2 - 11x -40 Add to -11 Multiply to -80 180 240 3--- 420 516 6--- 7 810 6.2

26. 10/16/2015Math 120 - KM26 2x 2 - 11x -40 Add to -11 Multiply to -80 6.2

27. 10/16/2015Math 120 - KM27 6x 2 - 17x +12 Add to -17 Multiply to 72 1 236 324 418 5-- 612 7-- 89 6.2

28. 10/16/2015Math 120 - KM28 Start with the GCF 6.2

29. 10/16/2015Math 120 - KM29 More Problems? 12y 3 + 22y 2 – 70y + 15x - 4x 2 - 9 5ax 3 + 20ax 2 – 160ax 2x 4 + 5x 2 + 12 2x 6 + 4x 3 – 30 6.2

30. 10/16/2015Math 120 - KM30 6.3 Special Factoring 6.3

31. 10/16/2015Math 120 - KM31 Special Factoring Shortcuts 6.3

32. 10/16/2015Math 120 - KM32 Special Polynomials 6.3

33. 10/16/2015Math 120 - KM33 Perfect Trinomial Square 6.3

34. 10/16/2015Math 120 - KM34 Perfect Trinomial Square 6.3

35. 10/16/2015Math 120 - KM35 Perfect Trinomial Square 6.3

36. 10/16/2015Math 120 - KM36 Perfect Trinomial Square 6.3

37. 10/16/2015Math 120 - KM37 OK – Short Cut Time! 6.3

38. 10/16/2015Math 120 - KM38 Difference of Squares 6.3

39. 10/16/2015Math 120 - KM39 You can do this! 6.3

40. 10/16/2015Math 120 - KM40 Check these out! 6.3

41. 10/16/2015Math 120 - KM41 Sum or Difference of Cubes nn cubed 11 28 327 464 5125 6216 …… nn3n3 6.3

42. 10/16/2015Math 120 - KM42 Sum or Difference of Cubes 6.3

43. 10/16/2015Math 120 - KM43 Sum or Difference of Cubes 6.3

44. 10/16/2015Math 120 - KM44 Sum or Difference of Cubes 6.3

45. 10/16/2015Math 120 - KM45 How about a harder one? 6.3

46. 10/16/2015Math 120 - KM46 6.4 Factoring: A General Strategy 6.4

47. 10/16/2015Math 120 - KM47 Factoring Strategy GCF 1) GREATEST COMMON FACTOR Check carefully to see if there is a GCF and factor it out. If the leading coefficient is negative, factor out -1. 6.4

48. 10/16/2015Math 120 - KM48 Factoring Strategy Number of Terms 2) Number of TERMS a) Four Terms: Try grouping b) Three Terms: i) a 2 + 2ab + b 2 Perfect Square ii) a 2 – 2ab + b 2 Perfect Square iii) ax 2 + bx + c UNFOIL c) Two Terms: i) a 2 - b 2 Difference of Squares ii) a 2 + b 2 Sum of Squares - NF iii) x 3 – y 3 Difference of Cubes iv) x 3 + y 3 Sum of Cubes 6.4

49. 10/16/2015Math 120 - KM49 Factor Completely: Example 1 2x 3 + 6x 2 – 8x - 24 = 2[ x 3 + 3x 2 – 4x – 12 ] = 2[ x 2 (x + 3) – 4(x+3) ] = 2[(x + 3)(x 2 – 4)] = 2[(x + 3)(x + 2)(x - 2)] = 2(x + 3)(x + 2)(x - 2) 6.4

50. 10/16/2015Math 120 - KM50 Factor Completely: Example 2 5x 3 - 80x 2 + 320x = 5x[ x 2 – 16x + 64 ] = 5x[(x - 8)(x - 8)] = 5x(x - 8) 2 6.4

51. 10/16/2015Math 120 - KM51 Factor Completely: Example 3 9x 2 + 12x - 5 12 -45 -315 = 9x 2 -3x + 15x - 5 = 3x(3x – 1) + 5(3x - 1) = (3x – 1)(3x + 5) 6.4

52. 10/16/2015Math 120 - KM52 Factor Completely: Example 4 125x 3 + 8y 3 = (5x + 2y)(25x 2 – 10xy + 4y 2 ) = (5x) 3 + (2y) 3 6.4

53. 10/16/2015Math 120 - KM53 Factor Completely: Example 5 x 2 + 10x – y 2 + 25 = x 2 + 10x + 25 – y 2 = (x + 5) 2 – y 2 = [(x + 5) + y] [(x + 5) - y] = (x + 5 + y)(x + 5 – y) 6.4

54. 10/16/2015Math 120 - KM54 4.8 Applications 6.4

55. 10/16/2015Math 120 - KM55 General Strategy for Solving Equations Using The Zero Factor Property 1) Arrange the equation so that one side is zero. 2) Completely factor the other side. 3) Set each factor equal to zero and solve, if possible. 4) Write the solution set. 5) Check each solution by substitution. 6.5

56. 10/16/2015Math 120 - KM56 Zero Factor Property Solve: 2x(x + 5)(x-3) = 0 6.5

57. 10/16/2015Math 120 - KM57 Zero Factor Property Solve: (2x - 7)(4x + 3)= 0 6.5

58. 10/16/2015Math 120 - KM58 Zero Factor Property Solve: 6x 2 = 3x Use the properties of equality to rearrange the terms of the equation so that it is equal to ZERO. 6.5

59. 10/16/2015Math 120 - KM59 Solve: x 2 = 169 or 6.5

60. 10/16/2015Math 120 - KM60 Solve: x 2 + 25 = 10x or 6.5

61. 10/16/2015Math 120 - KM61 Solve: 3x 2 = 2 - x or 6.5

62. 10/16/2015Math 120 - KM62 Solve: x + 12 = x(x – 3) or 6.5

63. 10/16/2015Math 120 - KM63 Solve: 2x 3 + 3x 2 = 18x + 27 or 6.5

64. 10/16/2015Math 120 - KM64 The Pool is Cool! Pat has a rectangular swimming pool. The length is 16 feet longer than the width. The surface area of the pool is 420 square feet. What are the dimensions of the pool? 6.5

65. 10/16/2015Math 120 - KM65 Let’s see a Diagram! w w + 16 Area = length x width 420 = (w+16)(w) w 2 + 16w – 420 = 0 (w - 14)(w + 30) = 0 w = 14 or w = -30 6.5

66. 10/16/2015Math 120 - KM66 Answer the Question! w w + 16 Pat’s pool is 14 feet wide and 30 feet long. = 14 feet = 14 + 16 = 30 feet 6.5

67. 10/16/2015Math 120 - KM67 Is it “Square”? Lilly and Mike are building a deck and want to make sure it is “square” (the corners are 90 degrees). If the deck is 12’ by 16’, what diagonal measurement is needed to be sure it is “square”? 12’ 16’ d 6.5

68. 10/16/2015Math 120 - KM68 Time for the Pythagorean Equation! 12’ 16’ d 6.5

69. 10/16/2015Math 120 - KM69 Solve for d d = -20 or d = 20 If the diagonal is 20’ long, the deck will be “square”. 6.5

70. 10/16/2015Math 120 - KM70 That’s All For Now!