Slide 1 / 128 Polynomials
Slide 2 / 128 Table of Contents Factors and GCF Factoring out GCF's Factoring Trinomials x 2 + bx + c Factoring Using Special Patterns Factoring Trinomials ax 2 + bx + c Factoring 4 Term Polynomials Mixed Factoring Solving Equations by Factoring
Slide 3 / 128 Factors and Greatest Common Factors Return to Table of Contents
Slide 4 / 128 Factors of 10 Factors of 15 Number Bank Factors Unique to 10 Factors 10 and 15 have in common Factors Unique to 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 What is the greatest common factor (GCF) of 10 and 15?
Slide 5 / 128 Factors of 12 Factors of 18 Number Bank Factors Unique to 12 Factors 12 and 18 have in common Factors Unique to 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 What is the greatest common factor (GCF) of 12 and 18?
Slide 6 / 128 1 What is the GCF of 12 and 15?
Slide 7 / 128 2 What is the GCF of 24 and 48?
Slide 8 / 128 3 What is the GCF of 72 and 54?
Slide 9 / 128 4 What is the GCF of 70 and 99?
Slide 10 / 128 5 What is the GCF of 28, 56 and 42?
Slide 11 / 128 Variables also have a GCF. The GCF of variables is the variable(s) that is in each term raised to the lowest exponent given. Example: Find the GCF and and and and and and
Slide 12 / 128 6 What is the GCF of and? A B C D
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Slide 16 / 128 Factoring out GCFs Return to Table of Contents
Slide 17 / 128 The first step in factoring is to determine its greatest monomial factor. If there is a greatest monomial factor other than 1, use the distributive property to rewrite the given polynomial as the product of this greatest monomial factor and a polynomial. Example 1 Factor each polynomial. a) 6x 4-15x 3 + 3x 2 Find the GCF 3x 2 6x 4 15x 3 3x 2 GCF: 3x 2 3x 2 3x 2 3x 2 Reduce each term of the polynomial dividing by the GCF 3x 2 (2x 2-5x + 1)
Slide 18 / 128 The first step in factoring is to determine its greatest monomial factor. If there is a greatest monomial factor other than 1, use the distributive property to rewrite the given polynomial as the product of this greatest monomial factor and a polynomial. Example 1 Factor each polynomial. b) 4m 3 n - 7m 2 n 2 Find the GCF GCF: m 2 n Reduce each term of the polynomial dividing by the GCF m 2 n(4n - 7n)
Slide 19 / 128 Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor. Example 2 Factor each polynomial. a) y(y - 3) + 7(y - 3) Find the GCF GCF: y - 3 (y - 3) ( y(y - 3) (y - 3) + 7(y - 3) (y - 3) ( Reduce each term of the polynomial dividing by the GCF (y - 3)(y + 7)
Slide 20 / 128 Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor. Example 2 Factor each polynomial. b) Find the GCF GCF: Reduce each term of the polynomial dividing by the GCF
Slide 21 / 128 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 = -(y - x)
Slide 22 / 128 10 True or False: y - 7 = -1( 7 + y) True False
Slide 23 / 128 11 True or False: 8 - d = -1( d + 8) True False
Slide 24 / 128 12 True or False: 8c - h = -1( -8c + h) True False
Slide 25 / 128 13 True or False: -a - b = -1( a + b) True False
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Slide 28 / 128 14 If possible, Factor A B C D Already Simplified
Slide 29 / 128 15 If possible, Factor A B C D Already Simplified
Slide 30 / 128 16 If possible, Factor A B C D Already Simplified
Slide 31 / 128 17 If possible, Factor A B C D Already Simplified
Slide 32 / 128 18 If possible, Factor A B C D Already Simplified
Slide 33 / 128 19 If possible, Factor A B C D Already Simplified
Slide 34 / 128 Factoring Trinomials: x 2 + bx + c Return to Table of Contents
Slide 35 / 128 A polynomial that can be simplified to the form ax + bx + c, where a 0, is called a quadratic polynomial. Quadratic term. Linear term. Constant term.
Slide 36 / 128 A quadratic polynomial in which b 0 and c 0 is called a quadratic trinomial. If only b=0 or c=0 it is called a quadratic binomial. If both b=0 and c=0 it is a quadratic monomial. Examples: Choose all of the description that apply. Cubic Quadratic Linear Constant Trinomial Binomial Monomial
Slide 37 / 128 20 Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial
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Slide 39 / 128 22 Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial
Slide 40 / 128 23 Choose all of the descriptions that apply to: A B C D E F Quadratic Linear Constant Trinomial Binomial Monomial
Slide 41 / 128 Simplify. 1) (x + 2)(x + 3) = 2) (x - 4)(x - 1) = 3) (x + 1)(x - 5) = 4) (x + 6)(x - 2) = Answer Bank x 2-5x + 4 x 2 + 5x + 6 x 2-4x - 5 x 2 + 4x - 12 Slide each polynomial from the circle to the correct expression. RECALL What did we do?? Look for a pattern!!
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Slide 44 / 128 Examples: (x - 8)(x - 1)
Slide 45 / 128 24 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
Slide 46 / 128 25 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
Slide 47 / 128 26 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)
Slide 48 / 128 27 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)
Slide 49 / 128 28 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)
Slide 50 / 128 29 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)
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Examples Slide 53 / 128
Slide 54 / 128 30 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
Slide 55 / 128 31 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
Slide 56 / 128 32 Factor x 2-4x - 12 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)
Slide 57 / 128 33 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)
Slide 58 / 128 34 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)
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Slide 60 / 128 Mixed Practice
Slide 61 / 128 36 Factor the following A (x - 2)(x - 4) B (x + 2)(x + 4) C (x - 2)(x +4) D (x + 2)(x - 4)
Slide 62 / 128 37 Factor the following A (x - 3)(x - 5) B (x + 3)(x + 5) C (x - 3)(x +5) D (x + 3)(x - 5)
Slide 63 / 128 38 Factor the following A (x - 3)(x - 4) B (x + 3)(x + 4) C (x +2)(x +6) D (x + 1)(x+12)
Slide 64 / 128 39 Factor the following A (x - 2)(x - 5) B (x + 2)(x + 5) C (x - 2)(x +5) D (x + 2)(x - 5)
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Slide 68 / 128 Factor: Factor out STEP 1 STEP 2 STEP 3 STEP 4
Slide 69 / 128 40 Factor completely: A B C D
Slide 70 / 128 41 Factor completely: A B C D
Slide 71 / 128 42 Factor completely: A B C D
Slide 72 / 128 43 Factor completely: A B C D
Slide 73 / 128 44 Factor completely: A B C D
Slide 74 / 128 Factoring Using Special Patterns Return to Table of Contents
Slide 75 / 128 When we were multiplying polynomials we had special patterns. Square of Sums Difference of Sums Product of a Sum and a Difference If we learn to recognize these squares and products we can use them to help us factor.
Slide 76 / 128 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: Recall: Observe the trinomial The first term is a perfect square. The second term is 2 times square root of the first term times the square root of the third. The sign is plus/minus. The third term is a perfect square.
Slide 77 / 128 Examples of Perfect Square Trinomials
Slide 78 / 128 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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Slide 80 / 128 45 Factor A B C D Not a perfect Square Trinomial
Slide 81 / 128 46 Factor A B C D Not a perfect Square Trinomial
Slide 82 / 128 47 Factor A B C D Not a perfect Square Trinomial
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Slide 84 / 128 Examples of Difference of Squares
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Slide 87 / 128 48 Factor A B C D Not a Difference of Squares
Slide 88 / 128 49 Factor A B C D Not a Difference of Squares
Slide 89 / 128 50 Factor A B C D Not a Difference of Squares
Slide 90 / 128 51 Factor using Difference of Squares: A B C D Not a Difference of Squares
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Slide 92 / 128 Factoring Trinomials: ax 2 + bx + c Return to Table of Contents
Slide 93 / 128 How to factor a trinomial of the form ax² + bx + c. Example: Factor 2d² + 15d + 18 Find the product of a and c : 2 18 = 36 Now find two integers whose product is 36 and whose sum is equal to b or 15. Factors of 36 Sum = 15? 1, 36 2, 18 3, 12 1 + 36 = 37 2 + 18 = 20 3 + 12 = 15 Now substitute 12 + 3 into the equation for 15. 2d² + (12 + 3)d + 18 Distribute 2d² + 12d + 3d + 18 Group and factor GCF 2d(d + 6) + 3(d + 6) Factor common binomial (d + 6)(2d + 3) Remember to check using FOIL!
Slide 94 / 128 Factor. 15x² - 13x + 2 a = 15 and c = 2, but b = -13 Since both a and c are 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, -6-1 + -30 = -31-2 + -15 = -17-3 + -10 = -13-5 + -6 = -11
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Factor 6y² - 13y - 5 Slide 96 / 128
Slide 97 / 128 A polynomial that cannot be written as a product of two polynomials is called a prime polynomial.
Slide 98 / 128 53 Factor A B C D Prime Polynomial
Slide 99 / 128 54 Factor A B C D Prime Polynomial
Slide 100 / 128 55 Factor A B C D Prime Polynomial
Slide 101 / 128 Factoring 4 Term Polynomials Return to Table of Contents
Slide 102 / 128 Polynomials with four terms like ab - 4b + 6a - 24, can 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 proper b(a - 4) + 6(a - 4) Factor the GCF (a - 4) (b + 6) Notice that a - 4 is a common binomial factor and factor!
Slide 103 / 128 Example 2: 6xy + 8x - 21y - 28 (6xy + 8x) + (-21y - 28) Group 2x(3y + 4) + (-7)(3y + 4) Factor GCF (3y +4) (2x - 7) Factor common binomial
Slide 104 / 128 You must be able to recognize additive inverses!!! (3 - a and a - 3 are additive inverses because their sum is equal to Remember 3 - a = -1(a - 3). Example 3: 15x - 3xy + 4y - 20 (15x - 3xy) + (4y - 20) Group 3x(5 - y) + 4(y - 5) Factor GCF 3x(-1)(-5 + y) + 4(y - 5) Notice additive inverses -3x(y - 5) + 4(y - 5) Simplify (y - 5) (-3x + 4) Factor common binomial Remember to check each problem by using FOIL.
Slide 105 / 128 56 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)
Slide 106 / 128 57 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)
Slide 107 / 128 58 Factor 20ab - 35b - 63 +36a A (4a - 7)(5b - 9) B (4a - 7)(5b + 9) C (4a + 7)(5b - 9) D (4a + 7)(5b + 9)
Slide 108 / 128 59 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)
Slide 109 / 128 Mixed Factoring Return to Table of Contents
Slide 110 / 128 Summary of Factoring Factor the Polynomial 2 Terms Difference of Squares Perfect Square Trinomial Factor out GCF 3 Terms Factor the Trinomial 4 Terms 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.
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Slide 112 / 128 60 Factor completely: A B C D
Slide 113 / 128 61 Factor completely A B C D prime polynomial
Slide 114 / 128 62 Factor A B C D prime polynomial
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Slide 116 / 128 64 Factor A B C D Prime Polynomial
Slide 117 / 128 Solving Equations by Factoring Return to Table of Contents
Slide 118 / 128 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. This is known as the Zero Product Property.
Slide 119 / 128 Given the following equation, what conclusion(s) can be dr (x - 4)(x + 3) = 0 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 = 0 + 4 + 4-3 - 3 x = 4 or x = -3 Therefore, our solution set is {-3, 4}. To verify the results, substitute solution back into the original equation. To check x = -3: (x - 4)(x + 3) = 0 To check x = 4: (x - 4)(x + 3) = 0 (-3-4)(-3 + 3) = 0 (-7)(0) = 0 0 = 0 (4-4)(4 + 3) = 0 (0)(7) = 0 0 = 0
Slide 120 / 128 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? Fac 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}
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Slide 124 / 128 65 Choose all of the solutions to: A B C D E F
Slide 125 / 128 66 Choose all of the solutions to: A B C D E F
Slide 126 / 128 67 Choose all of the solutions to: A B C D E F
Slide 127 / 128 68 A ball is thrown with its height at any time given by A B C D When does the ball hit the ground? -1 seconds 0 seconds 9 seconds 10 seconds
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