# Algebra I. Polynomials.

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2 Algebra I Polynomials

3 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying a Polynomial by a Polynomial Special Binomial Products Solving Equations Factors and GCF Factoring out GCF's Factoring Using Special Patterns Identifying & Factoring x 2 + bx + c Factoring Trinomials ax 2 + bx + c Factoring 4 Term Polynomials Mixed Factoring Solving Equations by Factoring Click on the topic to go to that section 3

5 Monomial 81y 4 z A monomial is a one term expression formed by a number, a variable, or the product of numbers and variables. Examples of monomials... rt 6 32,457 17x 2 4x 28 mn 3 We usually write the variables in exponential form exponents must be whole numbers. 5

6 Monomials Drag the following terms into the correct sorting box. If you sort correctly, the term will be visible. If you sort incorrectly, the term will disappear. x 2 (5 + 7y) 7x 3 y 5-4 t a + b -5 xy 4 7 5x rs 15 48x 2 yz 3 4(5a 2 bc 2 ) 6

7 A polynomial is an expression that contains one or more monomials. Examples of polynomials... 5a 2 Polynomials 7+b+c 2 +4d 3 rt 8x 3 +x 2 c 2 +d 6 + a 4 b 15 8a 3-2b 2 4c-mn 3 7

8 Polynomials What polynomials DO have: One or more terms made up of... Numbers Variables raised to wholenumber exponents Products of numbers and variables What polynomials DON'T have: Square roots of variables Negative exponents Fractional exponents Variables in the denominators of any fractions 8

9 Polynomials What is the exponent of the variable in the expression 5x? What is the exponent of the variable in the expression 5? Teacher 9

10 Degrees of Monomials The degree of a monomial is the sum of the exponents of its variables. The degree of a nonzero constant such as 5 or 12 is 0. The constant 0 has no degree. Examples: 1) The degree of 3x is? 2) The degree of 6x3y is? 3) The degree of 9 is? 10

11 1 What is the degree of x 2? A 0 B 1 C 2 D 3 11

12 2 What is the degree of mn? A 0 B 1 C 2 D 3 12

13 3 What is the degree of 3? A 0 B 1 C 2 D 3 13

14 4 What is the degree of 7t 8? 14

15 Degrees of Polynomials The degree of a polynomial is the same as that of the term with the greatest degree. Example: Find degree of the polynomial 4x 3 y 2 6xy 2 + xy. 4x 3 y 2 has a degree of 5, 6xy 2 has a degree of 3, xy has a degree of 2. The highest degree is 5, so the degree of the polynomial is 5. 15

16 Find the degree of each polynomial 1) 3 2) 12c 3 3) ab 4) 8s 4 t 5) 2 7n 6) h 4 8t 7) s 3 + 2v 2 y

17 5 What is the degree of the following polynomial: a 2 b 2 + c 4 d x 2 y A 3 B 4 C 5 D 6 17

18 6 What is the degree of the following polynomial: a 3 b 3 + c 4 d x 3 y 2 A 3 B 4 C 5 D 6 18

20 Standard Form A polynomial is in standard form when all of the terms are in order from highest degree to the lowest degree. Standard form is commonly accepted way to write polynomials. Example: 9x 7 8x x 4 3x 2 +2x 1 is in standard form. Drag each term to put the following equation into standard form: 67 11x 4 21x 9 9x 4 x 8 + 2x 3 x 20

21 Vocabulary Monomials with the same variables and the same power are like terms. The number in front of each term is called coefficient the of the term. If there is no variable in the term, the term is called constant the term. Like Terms Unlike Terms 4x and 12x 3b and 3a x 3 y and 4x 3 y 6a 2 b and 2ab 2 21

22 Like Terms Like terms can be combined by adding the coefficients, but keeping the variables the same. WHY? 3x + 5x means 3 times a number x added to 5 times the same number x. So altogether, we have 8 times the number x. What we are really doing is the distributive property of multiplication over addition in reverse: 3x + 5x = (3+5)x = 8x One big mistake students often make is to multiply the variables: 3x + 5x = 8x 2 22

23 Like Terms Combine these like terms using the indicated operation. 23

24 7 Simplify A B C D 24

25 8 Simplify A B C D 25

26 9 Simplify A B C D 26

27 Add Polynomials To add polynomials, combine the like terms from each polynomial. To add vertically, first line up the like terms and then add. Examples: (3x 2 +5x 12) + (5x 2 7x +3) (3x 4 5x) + (7x 4 +5x 2 14x) line up the like terms line up the like terms 3x 2 + 5x 12 3x 4 5x (+) 5x 2 7x + 3 (+) 7x 4 + 5x 2 14x 8x 2 2x 9 10x 4 + 5x 2 19x click click 27

28 Add Polynomials We can also add polynomials horizontally. (3x x 5) + (5x 2 7x 9) Use the communitive and associative properties to group like terms. (3x 2 + 5x 2 ) + (12x + 7x) + ( 5 + 9) click 8x 2 + 5x 14 28

29 10 Add A B C D 29

30 11 Add A B C D 30

31 12 Add A B C D 31

32 13 Add A B C D 32

33 14 Add A B C D 33

34 Subtract Polynomials To subtract polynomials, subtract the coefficients of like terms. Example: 3x 4x = 7x 13y ( 9y) = 22y 6xy 13xy = 7xy 34

35 15 Subtract A B C D 35

36 16 Subtract A B C D 36

37 Subtract Polynomials We can subtract polynomials vertically. To subtract a polynomial, change the subtraction to adding 1. Distribute the 1 and then follow the rules for adding polynomials (3x 2 +4x 5) (5x 2 6x +3) (3x 2 +4x 5) +( 1) (5x 2 6x+3) (3x 2 +4x 5) + ( 5x 2 +6x 3) click 3x 2 + 4x 5 (+) 5x 2 6x + 3 2x 2 +10x 8 37

38 Subtract Polynomials We can subtract polynomials vertically. Example: (4x 3 3x 5) (2x 3 +4x 2 7) (4x 3 3x 5) +( 1)(2x 3 +4x 2 7) (4x 3 3x 5) + ( 2x 3 4x 2 +7) 4x 3 3x 5 (+) 2x 3 4x x 3 4x 2 3x + 2 click 38

39 Subtract Polynomials We can also subtract polynomials horizontally. (3x x 5) (5x 2 7x 9) Change the subtraction to adding a negative one and distribute the negative one. (3x x 5) +( 1)(5x 2 7x 9) (3x x 5) + ( 5x 2 + 7x + 9) Use the communitive and associative properties to group like terms. (3x 2 + 5x 2 ) + (12x +7x) + ( 5 +9) 2x x + 4 click 39

40 17 Subtract A B C D 40

41 18 Subtract A B C D 41

42 19 Subtract A B C D 42

43 20 What is the perimeter of the following figure? (answers are in units) A B C D 43

44 Summary Is the sum or difference of two polynomials always a polynomial? When we add polynomials, we are adding the terms of the first to the terms of the second, and each of these sums is a new term of the same degree. Each new term consists of a constant times variables raised to whole number powers, so the sum is in fact a polynomial. Therefore, we say that the set of polynomials is "closed under addition". Since subtraction is just adding the opposite, the set of polynomials is also closed under subtraction. 44

46 Multiplying Polynomials Find the total area of the rectangles square units 46

47 Multiplying Polynomials Find the total area of the rectangles. square units 47

48 To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication. Example: 2x(5x 2 6x + 8) ( 2x)(5x 2 ) + ( 2x)( 6x) + ( 2x)(8) 10x x 2 16x Multiplying Polynomials 48

49 Multiplying Polynomials Let's Try It! Multiply to simplify. 1. x(2x 3 4x 2 + 7x) 2. 4x 2 (5x 2 6x 3) 3. 3xy(4x 3 y 2 5x 2 y 3 + 8xy 4 ) 49

50 21 What is the area of the rectangle shown? A B C D x 2 x 2 + 2x

51 22 Multiply A 6x 2 + 8x 12 B 6x 2 + 8x 2 12 C D 6x 2 + 8x 2 12x 6x 3 + 8x 2 12x 51

52 23 Multiply A B C D 52

53 24 Find the area of a triangle (A= 1 / 2 bh) with a base of 4x and a height of 2x 8. (All answers are in square units.) A B C D 53

54 25 Rewrite the expression 3a(a + b 5) + 4( 2a + 2b) + b(a + 3b 7) to find the coefficients of each term. Enter the coefficients into the appropriate boxes. a 2 + b 2 + ab + a + b From PARCC EOY sample test calculator #9 54

56 26 Find the area of the rectangle in two different ways

57 Multiply Polynomials To multiply a polynomial by a polynomial, you multiply each term of the first polynomial by each term of the second. Then, add like terms. Example 1: (2x + 4y)(3x + 2y) Example 2: (x + 3)(x2 + 2x + 4) 57

58 The FOIL Method is a shortcut that can be used to remember how multiply two binomials. To multiply two binomials, find the sum of the products of the... First terms of each binomial Outer terms the terms on the outsides Inner Terms the terms on the inside Last Terms of each binomial FOIL Method (a + b)(c + d) = ac + ad + bc + bd Remember FOIL is just a mnemonic to help you remember the steps for binomials. What you are really doing is multiplying each term in the first binomial by each term in the second. 58

59 Multiply Polynomials Try it!find each product. 1) (x 4)(x 3) 2) (x + 2)(3x 8) 59

60 Multiply Polynomials Try it! Find each product. 3) (2x 3y)(4x + 5y) 4) (3x 6)(x 2 2x) 60

61 27 What is the total area of the rectangles shown? A B C D 2x 4 4x 5 61

62 28 Multiply: A B C D 62

63 29 Multiply: A B C D 63

64 30 Multiply: A B C D 64

65 31 Multiply: A B C D 65

66 32 Find the area of a square with a side of A B C D 66

67 33 What is the area of the rectangle (in square units)? A B C D 3x 2 + 5x + 2 3x 2 + 6x + 2 3x 2 6x + 2 3x 2 5x +2 67

68 34 Find the total area of the rectangles. x 3 2x 4 68

70 35 What is the area of the shaded region (in sq. units)? A B C D 11x 2 + 3x 8 7x 2 + 3x 9 7x 2 3x 10 11x 2 3x 8 70

71 36 What is the area of the shaded region (in square units)? A B C D 2x 2 2x 8 2x 2 4x 6 2x 2 10x 8 2x 2 6x 4 71

73 Square of a Sum (a + b) 2 = (a + b)(a + b) = a 2 + ab + ab + b 2 = a 2 + 2ab + b 2 a b a b a 2 ab ab b 2 Notice that there are two of the term ab! 73

74 Square of a Difference (a b) 2 = (a b)(a b) = a 2 ab ab + b 2 = a 2 2ab + b 2 a a a 2 b ab b ab + b 2 Notice that there are two of the term ab! 74

75 Product of a Sum and a Difference (a + b)(a b) = a 2 ab + ab + b 2 a a b a 2 ab = a 2 b 2 + b + ab b 2 This time, the + ab and the ab add up to 0, and so the middle term drops out. 75

76 Special Products Try It! Find each product. 1) (3p + 9) 2 2) (6 p) 2 3) (2x 3)(2x + 3) 76

77 Fill in the missing pieces (3x - 5y) 2 = x 2 + xy + y 2 ( x + y) 2 = 9x 2 + xy + 36y 2 ( x + y) 2 = 121x 2-66xy + y 2 ( 12x - y)( x + 9y) = x 2 - y 2 77

78 37 (x 5) 2 A x B x x + 25 C x 2 10x + 25 D x

79 38 A B C D 79

80 39 What is the area of a square with sides 2x + 4? A B C D 80

81 40 A B C D 81

83 Zero Product Property Given the following equation, what conclusion(s) can be drawn? ab = 0 Since the product is 0, one of the factors, a or b, must be 0. 83

84 Zero Product Property If ab = 0, then either a = 0 or b = 0. Think about it: if 3x = 0, then what is x? 84

85 Zero Product Property What about this? (x 4)(x + 3) = 0 Since (x 4) is being multiplied by (x + 3), then each binomial is a FACTOR of the left side of the equation. Since the product is 0, one of the factors must be 0. Therefore, either x 4 = 0 or x + 3 = 0. x 4 = 0 or x + 3 = x = 4 or x = 3 85

86 Zero Product Property Therefore, our solution set is { 3, 4}. To verify the results, substitute each solution back into the original equation. To check x = 3: (x 4)(x + 3) = 0 ( 3 4)( 3 + 3) = 0 ( 7)(0) = 0 0 = 0 To check x = 4: (x 4)(x + 3) = 0 (4 4)(4 + 3) = 0 (0)(7) = 0 0 = 0 86

87 Solve What if you were given the following equation? (x 6)(x + 4) = 0 87

88 41 Solve (a + 3)(a 6) = 0. A {3, 6} B { 3, 6} C { 3, 6} D {3, 6} 88

89 42 Solve (a 2)(a 4) = 0. A {2, 4} B { 2, 4} C { 2, 4} D {2, 4} 89

90 43 Solve (2a 8)(a + 1) = 0. A { 1, 16} B { 1, 16} C { 1, 4} D { 1, 4} 90

92 GCF What is the greatest common factor (GCF) of 10 and 15? Factors of 10 Factors of 15 Number Bank Factors Unique to 10 Factors 10 and 15 have in common Factors Unique to

93 GCF What is the greatest common factor (GCF) of 12 and 18? Factors of 12 Factors of 18 Number Bank Factors Unique to 12 Factors 12 and 18 have in common Factors Unique to

94 44 What is the GCF of 12 and 15? 94

95 45 What is the GCF of 24 and 48? 95

96 46 What is the GCF of 72 and 54? 96

97 47 What is the GCF of 28, 56 and 42? 97

98 GCF Variables also have a GCF. The GCF of variables is the variable(s) that is in each term raised to the least exponent given. Example: Find the GCF x 2 and x 3 r 4, r 5 and r 8 x 3 y 2 and x 2 y 3 20x 2 y 2 z 5 and 15x 4 y 4 z 4 98

99 48 What is the GCF of and? A B C D 99

100 49 What is the GCF of and? A B C D 100

101 50 What is the GCF of and? and A B C D 101

102 51 What is the GCF of and and? A B C D 102

104 Factoring Factoring a number means to find other numbers you can multiply to get the number. 48 = 6 8, so 6 and 8 are both factors of 48. Factoring a polynomial means to find other polynomials that can be multiplied to get the original polynomial. (y + 1)(y 4) = y 2 3y 4, so y + 1, and y 4 are factors of y 2 3y

105 Example: Factoring Factor 10x 2 30x We might notice quickly that both terms have 10 as a factor, so we could have 10(x 2 3x). But both terms also have x as a factor. So the greatest common factor of both terms is 10x. 10x 2 30x = 10x (x 3) The left side of the equation is in expanded form, and the right side is in factored form. 105

106 Factoring The first step in factoring is to look for the greatest monomial factor. If there is a greatest monomial factor other than 1, use the distributive property in reverse to rewrite the given polynomial as the product of this greatest monomial factor and a polynomial. Example Factor 6x 4 15x 3 + 3x 2 106

107 Factoring Factor: 4m 3 n 7m 2 n 2 100x 5 20x x x x 107

108 Sometimes we can factor a polynomial that is not in simplest form but has a common binomial factor. Consider this problem: y(y 3) + 7(y 3) Factoring In this case, y 3 is the common factor. If we divide out the y 3's we get: (y 3) ( ) = (y 3)(y + 7) y(y 3) + 7(y 3) 108

109 Factoring Factor each polynomial: a(z 2 + 5) (z 2 + 5) 3x(x + y) + 4y(x + y) 7mn(x y) 2(x + y) 109

110 Factoring In working with common binomial factors, look for factors that are opposites of each other. For example: (x y) = (y x) because x y = x + ( y) = y + x = 1(y x) so x y and y x are opposites or additive inverses of each other. You can check this by adding them together: x y + y x = 0! 110

111 Additive Inverse Name the additive inverse of each binomial: 3x 1 5a + 3b x + y 4x 6y Prove that each pair are additive inverses by adding them together what do you get? 111

112 52 True or False: y 7 = 7 y True False 112

113 53 True or False: 8 d = 1( d + 8) True False 113

114 54 True or False: The additive inverse of 8c h is 8c + h. True False 114

115 55 True or False: a b and a + b are opposites. True False 115

116 Opposites In working with common binomial factors, look for factors that are opposites of each other. Example 3 Factor the polynomial. n(n 3) 7(3 n) Rewrite 3 n as 1(n 3) n(n 3) 7( 1)(n 3) Simplify n(n 3) + 7(n 3) Factor (n 3)(n + 7) 116

117 Factor the polynomial. p(h 1) + 4(1 h) 117

118 56 If possible, Factor A B C D Already Simplified 118

119 57 If possible, Factor A B C D Already Simplified 119

120 58 If possible, Factor A B C D Already Simplified 120

121 59 If possible, Factor A B C D Already Simplified 121

123 When we were multiplying polynomials we had special patterns. Square of Sums Difference of Sums Special Patterns in Multiplying Product of a Sum and a Difference If we learn to recognize these squares and products we can use them to help us factor. 123

124 Perfect Square Trinomials The Square of a Sum and the Square of a difference have products that are called Perfect Square Trinomials. How to Recognize a Perfect Square Trinomial: ( + ) 2 = ( ) 2 = Fill in the blanks with any monomial (or any expression!!) Try it!! 124

125 Perfect Square Trinomials Examples: 2 What do these trinomials have in common? What patterns do you see? 125

126 Perfect Square Trinomials Complete these perfect square equations: (x + ) 2 = x (x ) 2 = x (x 10) 2 = x (2x + ) 2 = x

127 Perfect Square Trinomials Is the trinomial a perfect square? Drag the Perfect Square Trinomials into the Box. Only Perfect Square Trinomials will remain visible. 127

128 Factoring Perfect Square Trinomials Once a Perfect Square Trinomial has been identified, it factors following the form: Examples: ( sq rt of ) 2 the last sq rt of the first term sign of the middle term term click click click 128

129 60 Factor A B C D Not a perfect Square Trinomial 129

130 61 Factor A B C D Not a perfect Square Trinomial 130

131 62 Factor A B C D Not a perfect Square Trinomial 131

132 Difference of Squares Binomials The product of a sum and difference of two monomials has a product called a Difference of Squares. How to Recognize a Difference of Squares Binomial: ( + )( )= 2 2 What happens to the middle term? Fill in the blanks with any monomial (or any expression!!) Try it!! 132

133 Difference of Squares Examples: 133

134 Difference of Squares Is the binomial a difference of squares? Drag the Difference of Squares binomials into the Box. Only Difference of Squares will remain visible. 134

135 Factoring a Difference of Squares Once a binomial is determined to be a Difference of Squares, it factors following the pattern: ( )( + ) sq rt of sq rt of sq rt of sq rt of 1 st term 2 nd term 1st term 2 nd term Factor each of the following: x y 2 4m 2 36n 2 y

136 63 Factor A B C D Not a Difference of Squares 136

137 64 Factor A B C D Not a Difference of Squares 137

138 65 Factor A B C D Not a Difference of Squares 138

139 66 Factor using Difference of Squares: A B C D Not a Difference of Squares 139

140 67 Factor A B C D 140

142 Classifying Polynomials Polynomials can be classified by the number of terms. The table below summarizes these classifications. 142

143 Classifying Polynomials Polynomials can be desribed based on something called their "degree". For a polynomial with one variable, the degree is the largest exponent of the variable. the degree of this polynomial is 7 3x 7 5x 4 + 8x 1 143

144 Classifying Polynomials Polynomials can also be classified by degree. The table below summarizes these classifications. 144

145 Classifying Polynomials Classify each polynomial based on the number of terms and its degree. Cubic Quadratic Linear Constant Trinomial Binomial Monomial 145

146 68 Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial 146

147 69 Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial 147

148 70 Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial 148

149 71 Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial 149

150 Simplify 1) (x + 2)(x + 3) = 2) (x 4)(x 1) = 3) (x + 1)(x 5) = 4) (x + 6)(x 2) = Bank x 2 5x + 4 x 2 4x 5 x 2 + 5x + 6 x 2 + 4x 12 RECALL What did we do?? Look for a pattern!! Slide each polynomial from the circle to the correct expression. 150

151 Multiply: (x + 3)(x +4) (x +3)(x 4) (x 3)(x + 4) (x 3)(x 4) What is the same and what is different about each product? What patterns do you see? What generalizations can be made about multiplication of binomials? Work in your groups to make a list and then share with the class. Make up your own example like the one above. Do your generalizations hold up? 151

152 Factor a Trinomial with a Lead Coefficient of 1 Recognize the pattern: Factors of 6 Sum to 5? 1, 6 7 Look for factors of 6 that have the same signs. Which pair of factors adds to +5? (Both factors must be positive.) 2,

153 Factor a Trinomial with a Lead Coefficient of 1 Recognize the pattern: Factors of 6 Sum to 7? 1, 6 7 Look for factors of 6 that have the same signs. Which pair of factors adds to 7? (Both factors must be negative.) 2,

154 Examples: Factor 154

155 Examples: Factor 155

156 72 What kind of signs will the factors of 12 have, given the following equation? A B C D Both positive Both Negative Bigger factor positive, the other negative The bigger factor negative, the other positive 156

157 73 The factors of 12 will have what kind of signs given the following equation? A B C D Both positive Both negative Bigger factor positive, the other negative The bigger factor negative, the other positive 157

158 74 Factor A (x + 12)(x + 1) B (x + 6)(x + 2) C (x + 4)(x + 3) D (x 12)(x 1) E (x 6)(x 1) F (x 4)(x 3) 158

159 75 Factor A (x + 12)(x + 1) B (x + 6)(x + 2) C (x + 4)(x + 3) D (x 12)(x 1) E (x 6)(x 1) F (x 4)(x 3) 159

160 76 Factor A (x + 12)(x + 1) B (x + 6)(x + 2) C (x + 4)(x + 3) D (x 12)(x 1) E (x 6)(x 1) F (x 4)(x 3) 160

161 77 Factor A (x + 12)(x + 1) B (x + 6)(x + 2) C (x + 4)(x + 3) D (x 12)(x 1) E (x 6)(x 2) F (x 4)(x 3) 161

162 Factor a Trinomial with a Lead Coefficient of 1 Recognize the pattern: Factors of 6 Sum to 5? 1, 6 5 2, 3 1 Look for factors of 6 that have opposite signs. Which pair of factors adds to 5. (The larger factor must be negative.) 162

163 Factor a Trinomial with a Lead Coefficient of 1 Recognize the pattern: Factors of 6 Sum to 1? 1, 6 5 2, 3 1 Look for factors of 6 that have opposite signs. Which pair of factors adds to 1? (The larger factor must be positive.) 163

164 Factor Examples 164

165 Factor Examples 165

166 78 The factors of 12 will have what kind of signs given the following equation? A B C D Both positive Both negative Bigger factor positive, the other negative The bigger factor negative, the other positive 166

167 79 The factors of 12 will have what kind of signs given the following equation? A B C D Both positive Both negative Bigger factor positive, the other negative The bigger factor negative, the other positive 167

168 80 Factor A (x + 12)(x 1) B (x + 6)(x 2) C (x + 4)(x 3) D (x 12)(x + 1) E (x 6)(x + 1) F (x 4)(x + 3) 168

169 81 Factor A (x + 12)(x 1) B (x + 6)(x 2) C (x + 4)(x 3) D (x 12)(x + 1) E (x 6)(x + 1) F unable to factor using this method 169

170 82 Factor A (x + 12)(x 1) B (x + 6)(x 2) C (x + 4)(x 3) D (x 12)(x + 1) E (x 6)(x + 1) F unable to factor using this method 170

171 Mixed Practice 171

172 83 Factor the following A (x 2)(x 4) B (x + 2)(x + 4) C (x 2)(x +4) D (x + 2)(x 4) 172

173 84 Factor the following A (x 3)(x 5) B (x + 3)(x + 5) C (x 3)(x +5) D (x + 3)(x 5) 173

174 85 Factor the following A (x 3)(x 4) B (x + 3)(x + 4) C (x +2)(x +6) D (x + 1)(x+12) 174

175 86 Factor the following A (x 2)(x 5) B (x + 2)(x + 5) C (x 2)(x +5) D (x + 2)(x 5) 175

177 How to factor a trinomial of the form ax² + bx + c. Example: Factor 2d² + 15d + 18 First, find ac: 2 18 = 36 a does not = 1 Now find two integers whose product is ac and whose sum is equal to b or 15. Factors of 36 Sum = 15? 1, 36 2, 18 3, = = =

178 a does not = 1 2d² + 15d + 18 ac = 36, b = 15 Our numbers: 3 and 12 Split the middle term, 15d, into 3d + 12d: 2d² + 3d + 12d + 18 first 2 terms last 2 terms Factor the first two terms and the last two terms: d(2d + 3) + 6(2d + 3) Factor out the common binomial (2d + 3)(d + 6) Remember to check by multiplying! 178

179 a does not = 1 Factor. 15x² 13x + 2 ac = 30, but b = 13 Since ac is positive, and b is negative we need to find two negative factors of 30 that add up to 13 Factors of 30 Sum = 13? 1, 30 2, 15 3, 10 5, = = = =

180 a does not = 1 15x² 13x + 2 ac = 30, b = 13 Our numbers: 3 and

181 a does not = 1 Factor. 2b 2 b 10 a = 2, c = 10, and b = 1 Since ac is negative, and b is negative we need to find two factors with opposite signs whose product is 20 and that add up to 1. Since b is negative, larger factor of 20 must be negative. Factors of 20 Sum = 1? 181

182 a does not = 1 Factor 6y² 13y 5 182

183 Step 1: Calculate ac. Berry Method to Factor Step 2: Find a pair of numbers m and n, whose product is ac, and whose sum is b. Step 3: Create the product (ax + m)(ax + n). Step 4: From each binomial in step 3, factor out and discard any common factor. The result is your factored form. Example: 4x 2 19x + 12 ac = 48, b = 19 m = 3, n = 16 (4x 3)(4x 16) Factor 4 out of 4x 16 and toss it! (4x 3)(x 4) THE ANSWER! 183

184 Prime Polynomial A polynomial that cannot be factored as a product of two polynomials is called a prime polynomial. How can you tell if a polynomial is prime? Discuss with your table. click to reveal If there are no two integers whose product is ac and whose sum is b. 184

185 87 Factor A B C D Prime Polynomial 185

186 88 Factor A B C D Prime Polynomial 186

187 89 Factor A B C D Prime Polynomial 187

189 4 Terms Polynomials with four terms like ab 4b + 6a 24, can sometimes be factored by grouping terms of the polynomials. Example 1: ab 4b + 6a 24 (ab 4b) + (6a 24) Group terms into binomials that can be factored using the distributive property b(a 4) + 6(a 4) Factor the GCF (a 4) (b + 6) 189

190 4 Terms Example 6xy + 8x 21y

191 What are the relationships among the following: Some are equivalent, some are opposites, some are not related at all. Mix and match by dragging pairs for each category: Equivalent Opposites Not related x + 3 x + 3 x 3 x 3 3 x 3 + x 191

192 Additive Inverses You must be able to recognize additive inverses!!! (3 a and a 3 are additive inverses because their sum is equal to zero.) Remember 3 a = 1(a 3). Example 15x 3xy + 4y 20 (15x 3xy) + (4y 20) Group 3x(5 y) + 4(y 5) Factor GCF 3x( 1)(y 5) + 4(y 5) Rewrite based on additive inverses 3x(y 5) + 4(y 5) Simplify (y 5) ( 3x + 4) Factor common binomial Remember to check each problem by using FOIL. 192

193 90 Factor 15ab 3a + 10b 2 A (5b 1)(3a + 2) B (5b + 1)(3a + 2) C (5b 1)(3a 2) D (5b + 1)(3a 1) 193

194 91 Factor 10m 2 n 25mn + 6m 15 A B C D (2m 5)(5mn 3) (2m 5)(5mn+3) (2m+5)(5mn 3) (2m+5)(5mn+3) 194

195 92 Factor 20ab 35b a A (4a 7)(5b 9) B (4a 7)(5b + 9) C (4a + 7)(5b 9) D (4a + 7)(5b + 9) 195

196 93 Factor a 2 ab + 7b 7a A (a b)(a 7) B (a b)(a + 7) C (a + b)(a 7) D (a + b)(a + 7) 196

198 Summary of Factoring Factor the Polynomial 2 Terms Factor out GCF 3 Terms 4 Terms Difference of Squares Perfect Square Trinomial Factor the Trinomial Group and Factor out GCF. Look for a Common Binomial a = 1 a = 1 Check each factor to see if it can be factored again. If a polynomial cannot be factored, then it is called prime. 198

199 Examples 3r 3 9r 2 + 6r 3r(r 2 3r + 2) 3r(r 1)(r 2) 199

200 94 Factor completely: A B C D 200

201 95 Factor completely A B C D prime polynomial 201

202 96 Factor A B C D prime polynomial 202

203 97 Factor completely 10w 2 x 2 100w 2 x +1000w 2 A 10w 2 (x + 10) 2 B 10w 2 (x 10) 2 C 10(wx 10) 2 D 10w 2 (x 2 10x +100) 203

204 98 Factor A B C D Prime Polynomial 204

206 Given the following equation, what conclusion(s) can be drawn? ab = 0 206

207 Recall ~ Given the following equation, what conclusion(s) can drawn? be (x 4)(x + 3) = 0 Since the product is 0, one of the factors must be 0. Therefore, eitherx 4 = 0 or x + 3 = 0. x 4 = 0 or x + 3 = x = 4 or x = 3 Therefore, our solution set is { 3, 4}. To verify the results, substitute each solution back into the original equation. To check x = 3: (x 4)(x + 3) = 0 ( 3 4)( 3 + 3) = 0 ( 7)(0) = 0 0 = 0 To check x = 4: (x 4)(x + 3) = 0 (4 4)(4 + 3) = 0 (0)(7) = 0 0 = 0 207

208 What if you were given the following equation? How would you solve it? We can use the Zero Product Property to solve it. How can we turn this polynomial into a multiplication problem? Factor it! Factoring yields: (x 6)(x + 4) = 0 By the Zero Product Property: x 6 = 0 or x + 4 = 0 After solving each equation, we arrive at our solution: { 4, 6} 208

209 Solve Trinomial Recall the Steps for Factoring a Trinomial 1) See if a monomial can be factored out. 2) Need 2 numbers that multiply to the constant 3) and add to the middle number. 4) Write out the factors. Now... 1) Set each binomial equal to zero. 2) Solve each binomial for the variable. click to reveal 209

210 Solve Trinomial Recall the Steps for Factoring a Trinomial 1) See if a monomial can be factored out. 2) Need 2 numbers that multiply to the constant 3) and add to the middle number. 4) Write out the factors. Now... 1) Set each binomial equal to zero. 2) Solve each binomial for the variable. click to reveal 210

211 Zero Product rule works only when the product of factors equals zero. If the equation equals some value other than zero, subtract to make one side of the equation zero. Example Zero Product Rule 211

212 99 Choose all of the solutions to: A B C D E F 212

213 100 Choose all of the solutions to: A 4 B 2 C 0 D 2 E 4 F

214 101 Choose all of the solutions to: A 4 B 2 C 0 D 2 E 4 F

215 Application A science class launches a toy rocket. The teacher tells the class that the height of the rocket at any given time is h = 16t t. When will the rocket hit the ground? When the rocket hits the ground, its height is 0. So h=0 which can be substituted into the equation: The rocket had to hit the ground some time after launching. The rocket hits the ground in 20 seconds. The 0 is an extraneous (extra) answer. 215

216 102 A ball is thrown with its height at any time given by When does the ball hit the ground? A B C D 1 seconds 0 seconds 9 seconds 10 seconds 216

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