# Algebra I Polynomials

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1 Slide 1 / 217

2 Slide 2 / 217 Algebra I Polynomials

3 Slide 3 / 217 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

4 Slide 4 / 217 Links to PARCC sample questions End of Year Calculator #9

6 Slide 6 / 217 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 4x 32, x 2 6 mn 3 We usually write the variables in exponential form - exponents must be whole numbers.

7 Slide 7 / 217 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 )

8 Slide 8 / 217 Polynomials A polynomial is an expression that contains one or more monomials. Examples of polynomials... 5a 2 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

9 Slide 9 / 217 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

10 Slide 10 / 217 Polynomials What is the exponent of the variable in the expression 5x? What is the exponent of the variable in the expression 5?

11 Slide 11 / 217 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?

12 Slide 12 / What is the degree of x 2? A 0 B 1 C 2 D 3

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

14 Slide 14 / What is the degree of 3? A 0 B 1 C 2 D 3

15 4 What is the degree of 7t 8? Slide 15 / 217

16 Slide 16 / 217 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.

17 Slide 17 / 217 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 2-1

18 Slide 18 / 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

19 Slide 19 / 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

21 Slide 21 / 217 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

22 Slide 22 / 217 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

23 Slide 23 / 217 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

24 Slide 24 / 217 Like Terms Combine these like terms using the indicated operation.

25 Slide 25 / Simplify A B C D

26 Slide 26 / Simplify A B C D

27 Slide 27 / Simplify A B C D

28 Slide 28 / 217 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 x 4-5x (+) 5x 2-7x + 3 (+) 7x 4 + 5x 2-14x 8x 2-2x x 4 + 5x 2-19x click click

29 Slide 29 / 217 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) + ( ) 8x 2 + 5x - 14

30 Slide 30 / Add A B C D

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35 Slide 35 / 217 Subtract Polynomials To subtract polynomials, subtract the coefficients of like terms. Example: -3x - 4x = -7x 13y - (-9y) = 22y 6xy - 13xy = -7xy

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38 Slide 38 / 217 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

39 Slide 39 / 217 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

40 Slide 40 / 217 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

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45 Slide 45 / 217 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.

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48 Slide 48 / 217 Multiplying Polynomials Find the total area of the rectangles. square units

49 Slide 49 / 217 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

50 Slide 50 / 217 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 )

51 Slide 51 / What is the area of the rectangle shown? A B C D x 2 x 2 + 2x + 4

52 Slide 52 / Multiply A 6x 2 + 8x - 12 B 6x 2 + 8x 2-12 C D 6x 2 + 8x 2-12x 6x 3 + 8x 2-12x

53 Slide 53 / Multiply A B C D

54 Slide 54 / 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

55 Slide 55 / Rewrite the expression -3a(a + b - 5) + 4(-2a + 2b) + b(a + 3b - 7) to find Students the type coefficients their answers hereof each term. Enter the coefficients into the appropriate boxes. a 2 + b 2 + ab + a + b From PARCC sample test

57 Slide 57 / Find the area of the rectangle in two different ways

58 Slide 58 / 217 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)

59 Slide 59 / 217 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.

60 Slide 60 / 217 Multiply Polynomials Try it! Find each product. 1) (x - 4)(x - 3) 2) (x + 2)(3x - 8)

61 Slide 61 / 217 Multiply Polynomials Try it! Find each product. 3) (2x - 3y)(4x + 5y) 4) (3x - 6)(x 2-2x)

62 Slide 62 / What is the total area of the rectangles shown? A B C D 2x 4 4x 5

63 Slide 63 / Multiply: A B C D

64 Slide 64 / Multiply: A B C D

65 Slide 65 / Multiply: A B C D

66 Slide 66 / Multiply: A B C D

67 Slide 67 / Find the area of a square with a side of A B C D

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69 Slide 69 / Find the total area of the rectangles. Students type their answers here x 2x 4 3

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74 Slide 74 / 217 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!

75 Slide 75 / 217 Square of a Difference (a - b) 2 = (a - b)(a - b) = a 2 - ab - ab + b 2 = a 2-2ab + b 2 a - b a a 2 - b - ab - ab + b 2 Notice that there are two of the term -ab!

76 Slide 76 / 217 Product of a Sum and a Difference (a + b)(a - b) a = a 2 - ab + ab + -b 2 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.

77 Slide 77 / 217 Special Products Try It! Find each product. 1) (3p + 9) 2 2) (6 - p) 2 3) (2x - 3)(2x + 3)

78 Slide 78 / 217 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

79 Slide 79 / (x - 5) 2 A x B x x + 25 C x 2-10x + 25 D x 2-25

80 Slide 80 / A B C D

81 Slide 81 / What is the area of a square with sides 2x + 4? A B C D

82 Slide 82 / A B C D

84 Slide 84 / 217 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.

85 Slide 85 / 217 Zero Product Property If ab = 0, then either a = 0 or b = 0. Think about it: if 3x = 0, then what is x?

86 Slide 86 / 217 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

87 Slide 87 / 217 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

88 Slide 88 / 217 Solve What if you were given the following equation? (x - 6)(x + 4) = 0

89 Slide 89 / Solve (a + 3)(a - 6) = 0. A {3, 6} B {-3, -6} C {-3, 6} D {3, -6}

90 Slide 90 / Solve (a - 2)(a - 4) = 0. A {2, 4} B {-2, -4} C {-2, 4} D {2, -4}

91 Slide 91 / Solve (2a - 8)(a + 1) = 0. A {-1, -16} B {-1, 16} C {-1, 4} D {-1, -4}

93 Slide 93 / 217 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

94 Slide 94 / 217 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

95 Slide 95 / What is the GCF of 12 and 15?

96 Slide 96 / What is the GCF of 24 and 48?

97 Slide 97 / What is the GCF of 72 and 54?

98 Slide 98 / What is the GCF of 28, 56 and 42?

99 Slide 99 / 217 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

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

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105 Slide 105 / 217 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 - 4.

106 Slide 106 / 217 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.

107 Slide 107 / 217 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

108 Slide 108 / 217 Factoring Factor: 4m 3 n - 7m 2 n 2 100x 5-20x x x x

109 Slide 109 / 217 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)

110 Slide 110 / 217 Factoring Factor each polynomial: a(z 2 + 5) - (z 2 + 5) 3x(x + y) + 4y(x + y) 7mn(x - y) - 2(x + y)

111 Slide 111 / 217 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!

112 Slide 112 / 217 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?

113 Slide 113 / True or False: y - 7 = y True False

114 Slide 114 / True or False: 8 - d = -1( d + 8) True False

115 Slide 115 / True or False: The additive inverse of 8c - h is -8c + h. True False

116 Slide 116 / True or False: -a - b and a + b are opposites. True False

117 Slide 117 / 217 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)

118 Slide 118 / 217 Factor the polynomial. p(h - 1) + 4(1 - h)

119 Slide 119 / If possible, Factor A B C D Already Simplified

120 Slide 120 / If possible, Factor A B C D Already Simplified

121 Slide 121 / If possible, Factor A B C D Already Simplified

122 Slide 122 / If possible, Factor A B C D Already Simplified

124 Slide 124 / 217 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.

125 Slide 125 / 217 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!!

126 Slide 126 / 217 Perfect Square Trinomials Examples: 2 What do these trinomials have in common? What patterns do you see?

127 Slide 127 / 217 Perfect Square Trinomials Complete these perfect square equations: (x + ) 2 = x (x - ) 2 = x (x - 10) 2 = x (2x + ) 2 = x

128 Slide 128 / 217 Perfect Square Trinomials Is the trinomial a perfect square? Drag the Perfect Square Trinomials into the Box. Only Perfect Square Trinomials will remain visible.

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130 Slide 130 / Factor A B C D Not a perfect Square Trinomial

131 Slide 131 / Factor A B C D Not a perfect Square Trinomial

132 Slide 132 / Factor A B C D Not a perfect Square Trinomial

133 Slide 133 / 217 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!!

134 Slide 134 / 217 Difference of Squares Examples:

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136 Slide 136 / 217 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 4-1

137 Slide 137 / Factor A B C D Not a Difference of Squares

138 Slide 138 / Factor A B C D Not a Difference of Squares

139 Slide 139 / Factor A B C D Not a Difference of Squares

140 Slide 140 / Factor using Difference of Squares: A B C D Not a Difference of Squares

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143 Slide 143 / 217 Classifying Polynomials Polynomials can be classified by the number of terms. The table below summarizes these classifications.

144 Slide 144 / 217 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

145 Slide 145 / 217 Classifying Polynomials Polynomials can also be classified by degree. The table below summarizes these classifications.

146 Slide 146 / 217 Classifying Polynomials Classify each polynomial based on the number of terms and its degree. Cubic Quadratic Linear Constant Trinomial Binomial Monomial

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

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

150 Slide 150 / Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial

151 Slide 151 / 217 Simplify 1) (x + 2)(x + 3) = Answer Bank 2) (x - 4)(x - 1) = 3) (x + 1)(x - 5) = 4) (x + 6)(x - 2) = x 2-5x + 4 x 2-4x - 5 x 2 + 5x + 6 x 2 + 4x - 12 Slide each polynomial from the circle to the correct expression. RECALL What did we do?? Look for a pattern!!

152 Slide 152 / 217 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?

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155 Slide 155 / 217 Examples: Factor

156 Slide 156 / 217 Examples: Factor

157 Slide 157 / 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

158 Slide 158 / 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

159 Slide 159 / 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 Slide 160 / 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)

161 Slide 161 / 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)

162 Slide 162 / 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)

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165 Slide 165 / 217 Factor Examples

166 Slide 166 / 217 Factor Examples

167 Slide 167 / 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

168 Slide 168 / 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

169 Slide 169 / 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)

170 Slide 170 / 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

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172 Slide 172 / 217 Mixed Practice

173 Slide 173 / Factor the following A (x - 2)(x - 4) B (x + 2)(x + 4) C (x - 2)(x +4) D (x + 2)(x - 4)

174 Slide 174 / Factor the following A (x - 3)(x - 5) B (x + 3)(x + 5) C (x - 3)(x +5) D (x + 3)(x - 5)

175 Slide 175 / Factor the following A (x - 3)(x - 4) B (x + 3)(x + 4) C (x +2)(x +6) D (x + 1)(x+12)

176 Slide 176 / Factor the following A (x - 2)(x - 5) B (x + 2)(x + 5) C (x - 2)(x +5) D (x + 2)(x - 5)

178 Slide 178 / 217 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, = = = 15

179 Slide 179 / 217 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!

180 Slide 180 / 217 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, = = = = -11

181 Slide 181 / 217 a does not = 1 15x² - 13x + 2 ac = 30, b = -13 Our numbers: -3 and -10

182 Slide 182 / 217 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?

183 Slide 183 / 217 a does not = 1 Factor 6y² - 13y - 5

184 Step 1: Calculate ac. Slide 184 / 217 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!

185 Slide 185 / 217 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.

186 Slide 186 / Factor A B C D Prime Polynomial

187 Slide 187 / Factor A B C D Prime Polynomial

188 Slide 188 / Factor A B C D Prime Polynomial

190 Slide 190 / 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) b(a - 4) + 6(a - 4) Group terms into binomials that can be factored using the distributive property Factor the GCF (a - 4) (b + 6)

191 Slide 191 / Terms Example 6xy + 8x - 21y - 28

192 Slide 192 / 217 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 x 3 + x

193 Slide 193 / 217 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.) Remember3 - 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.

194 Slide 194 / 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)

195 Slide 195 / 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)

196 Slide 196 / Factor 20ab - 35b a A (4a - 7)(5b - 9) B (4a - 7)(5b + 9) C (4a + 7)(5b - 9) D (4a + 7)(5b + 9)

197 Slide 197 / 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)

199 Slide 199 / 217 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.

200 Slide 200 / 217

201 Slide 201 / Factor completely: A B C D

202 Slide 202 / Factor completely A B C D prime polynomial

203 Slide 203 / Factor A B C D prime polynomial

204 Slide 204 / 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)

205 Slide 205 / Factor A B C D Prime Polynomial

207 Slide 207 / 217 Given the following equation, what conclusion(s) can be drawn? ab = 0

208 Slide 208 / 217 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

209 Slide 209 / 217 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}

210 Solve Slide 210 / 217 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

211 Slide 211 / 217

212 Slide 212 / 217

213 Slide 213 / Choose all of the solutions to: A B C D E F

214 Slide 214 / Choose all of the solutions to: A -4 B -2 C 0 D 2 E 4 F 16

215 Slide 215 / Choose all of the solutions to: A -4 B -2 C 0 D 2 E 4 F 16

216 Slide 216 / 217

217 Slide 217 / 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

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