1. Beyond FOIL Alternate Methods for Multiplying and Factoring Polynomials

2. FOIL Method Distributive Method Box Method Vertical Method Multiplying Polynomials

3. Distributive Method STEP 1: Rewrite the problem

5. Distributive Method STEP 2: Distribute

7. Distributive Method STEP 3: Combine Like Terms

9. Multiplying Polynomials (5x – 6)(3x + 8) WATCH THOSE SIGNS!!!

10. Rewrite the problem

11. Distribute

12. Combine Like Terms

13. Binomial x Trinomial Multiplying Polynomials

14. Rewrite the problem

15. Distribute

16. Combine Like Terms

17. (3x + 2)(5x + 4) Multiplying Polynomials BOX Method

18. STEP 1: Draw the BOX

19. Draw the Box 2x2 for a Binomial x Binomial

20. BOX Method STEP 2: Place terms on outside

22. BOX Method STEP 3: Multiply: Find the area of each box.

24. BOX Method STEP 3: Combine Like Terms

26. BOX Method LET’S SEE THAT AGAIN!

32. What about a binomial x trinomial?

35. Vertical Method

36. How do you multiply without a calculator?

38. What if we tried it this way?

41. Can we do that again?

44. MULTIPLYING POLYNOMIALS (3x + 2)(5x + 4)

45. VERTICAL Method STEP 1: Rewrite the Problem

46. VERTICAL Method

47. STEP 2: MULTIPLY

49. VERTICAL Method STEP 3: Combine Like Terms

51. VERTICAL Method

54. WHAT IF IT’S A TRINOMIAL x A BINOMIAL?

55. VERTICAL Method STEP 1: Rewrite the Problem

56. VERTICAL Method

57. STEP 2: MULTIPLY

59. VERTICAL Method STEP 3: Combine Like Terms

61. A SHORTCUT IS NOT A SHORTCUT IF IT IS THE ONLY WAY YOU KNOW.

63. FIRST FOIL METHOD F

64. OUTER FOIL METHOD O

65. INNER FOIL METHOD I

66. LAST FOIL METHOD L

68. Kinda

69. By Grouping GCF Trinomials Factoring Polynomials

70. Factor Pairs 24 1 · 24 2 · 12 3 · 8 4 · 6 40 1 · 40 2 · 20 4 · 10 5 · 8 84 1 · 84 2 · 42 3 · 28 4 · 21 6 · 14 7 · 12

71. Greatest Common Factor 63 1 · 63 3 · 21 7 · 9 84 1 · 84 2 · 42 3 · 28 4 · 21 6 · 14 7 · 12

72. ( ) Factor by Grouping 15x 2 + 12xy + 35xz + 28yz 3x ( 5x ) + 7z ( 5x ) + 4y (5x + 4y)(3x+ 7z)

73. ( ) Factor by Grouping 24ac – 9ad – 32bc + 12bd NEGATIVECHANGE -

74. ( ) Factor by Grouping 24ac – 9ad – 32bc + 12bd 3a ( 8c ) - 4b ( 8c ) - 3d (8c – 3d)(3a- 4b) -

75. Factoring Trinomials without a leading coefficient x 2 + 8x + 15

76. Factor Start Here Ask Yourself: What are the factor pairs of 15, 1 · 15 3 · 5

77. x 2 + 8x + 15 Factor Start Here Ask Yourself: What are the factor pairs of 15, 1 · 15 3 · 5 whose sum 1+= 16 3+= 8 is 8?

78. x 2 + 8x + 15 Factor 15 5 1+= 16 3+= 8 x( ) x35++ What signs would make a + 8?

79. x 2 + 5x - 24 Factor Start Here Ask Yourself: What are the factor pairs of 24, 1 · 24 2 · 12 3 · 8 4 · 6

80. x 2 + 5x - 24 Factor Start Here Ask Yourself: What are the factor pairs of 24, whose difference 1- 2- is 5? 1 · 24 2 · 12 3 · 8 4 · 6 3- 4- = 23 = 10 = 5 = 2

81. ( ) 4- 3- x 2 + 5x - 24 Factor 1- 2- 24 12 8 6 = 23 = 10 = 5 = 2 x( )x38-+ What signs would make a + 5?

82. x 2 – 8x - 105 Factor Start Here Ask Yourself: What are the factor pairs of 105, 1 · 105 3 · 35 5 · 21 7 · 15

83. 7- 5- 3 · 105 x 2 - 8x - 105 Factor Start Here Ask Yourself: What are the factor pairs of 105, whose difference 1- 3- is 8? 1 · 35 5 · 21 7 · 15 =104 = 32 = 16 = 8

84. 3- 7- 5- 105 x 2 - 8x - 105 Factor 1- 35 21 15 =104 = 32 = 16 = 8 ( )x x715+- What signs would make a - 8?

85. Factoring Trinomials with a leading coefficient 6x 2 + 19x + 10

86. Factor 1 st Step Multiply Leading Coefficient and Constant

87. Multiply 6x 2 + 19x + 10 60 x 2 nd Step Factor Pairs of 60

88. Factor Pairs 6x 2 + 19x + 10 60 1 · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 3 rd Step Whose sum Is 19. =61 =32 =23 =19

89. Rewrite 6x 2 + 19x + 10 60 1 · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 4 th Step Rewrite the Polynomial =61 =32 =23 =19

90. Rewrite 6x 2 + 19x + 10 60 1 · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 First Term =61 =32 =23 =19 6x 2 Factor Pair 4x15x Last Term + 10 Choose Signs ++

91. Rewrite 6x 2 + 19x + 10 60 1 · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 5 th Step Factor by Grouping =61 =32 =23 =19 6x 2 4x15x+ 10++

92. ( ) 2x 6x 2 ( ) Grouping 6x 2 + 19x + 10 60 1 · 60 2 · 30 3 · 20 4 · 15 5 · 12 6 · 10 =61 =32 =23 =19 4x15x+10++ 2x ( ) 3x+ 2 55 +5 ( ) 3x + 2 (3x + 2)(2x+5)

93. Factoring With the BOX x 2 – 10x + 16

94. x 2 -10x + 16 Factor Start Here Ask Yourself: What are the factor pairs of 16, 1 · 16 2 · 8 4 · 4

95. 4+ 2 · 1 · x 2 - 10x + 16 Factor Start Here Ask Yourself: What are the factor pairs of 24, whose sum 1+ 2+ is 10? 16 8 4 · 4 = 17 = 10 = 8

96. 1+ 2+ 4+ x 2 - 10x + 16 BOX 16 8 4 = 17 = 10 = 8 Place terms inside the box. x2x2 -2x -8x 16 Use signs to make -10x

97. 1+ 2+ 4+ x 2 - 10x + 16 BOX 16 8 4 = 17 = 10 = 8 Find the GCF of the columns and rows x2x2 -2x -8x 16 x-2 x -8

98. x2x2 -2x -8x 16 x-2 x -8 (x - 2)(x - 8)

99. Factoring With the BOX 3x 2 – 4x - 4

100. Multiply 3x 2 - 4x – 4 12 x

101. 3x 2 - 4x – 4 Factor Pairs 12 1 · 12 2 · 6 3 · 4 =11 =4 =1

102. 1- 2- 3- 3x 2 - 4x - 4 BOX 12 6 4 = 11 = 4 = 1 Place terms inside the box. 3x 2 2x -6x -4 Use signs to make -4x

103. 1- 2- 3- 3x 2 - 4x - 4 BOX 12 6 4 = 11 = 4 = 1 Find the GCF of the columns and rows 3x 2 2x -6x -4 3x2 x -2

104. 3x 2 2x -6x -4 3x2 x -2 (3x + 2)(x - 2)

105. Thank You!! Todd Rackowitz Independence High School todd.rackowitz@cms.k12.nc.us