Polynomials. Factors and Greatest Common Factors. Slide 1 / 128. Slide 2 / 128. Slide 3 / 128. Table of Contents

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1 Slide 1 / 128 Polynomials Table of ontents Slide 2 / 128 Factors and GF Factoring out GF'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 ommon Factors Return to Table of ontents

2 Factors of 10 Factors of 15 Number ank Slide 4 / 128 Factors Unique to 10 Factors 10 and 15 have in common Factors Unique to What is the greatest common factor (GF) of 10 and 15? Factors of 12 Factors of 18 Factors Unique to 12 Factors 12 and 18 have in common Factors Unique to 18 Number ank Slide 5 / 128 What is the greatest common factor (GF) of 12 and 18? 1 What is the GF of 12 and 15? Slide 6 / 128

3 2 What is the GF of 24 and 48? Slide 7 / What is the GF of 72 and 54? Slide 8 / What is the GF of 70 and 99? Slide 9 / 128

4 5 What is the GF of 28, 56 and 42? Slide 10 / 128 Variables also have a GF. Slide 11 / 128 The GF of variables is the variable(s) that is in each term raised to the lowest exponent given. Example: Find the GF and and and and and and 6 What is the GF of and? Slide 12 / 128

5 Slide 13 / 128 Slide 14 / 128 Slide 15 / 128

6 Slide 16 / 128 Factoring out GFs Return to Table of ontents 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. Slide 17 / 128 Example 1 Factor each polynomial. a) 6x 4-15x 3 + 3x 2 Find the GF 3x 2 6x 4 15x 3 3x 2 GF: 3x 2 3x 2 3x 2 3x 2 Reduce each term of the polynomial dividing by the GF 3x 2 (2x 2-5x + 1) 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. Slide 18 / 128 Example 1 Factor each polynomial. b) 4m 3 n - 7m 2 n 2 Find the GF GF: m 2 n Reduce each term of the polynomial dividing by the GF m 2 n(4n - 7n)

7 Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor. Slide 19 / 128 Example 2 Factor each polynomial. a) y(y - 3) + 7(y - 3) Find the GF GF: y - 3 Reduce each term of the polynomial dividing by the GF (y - 3) ( y(y - 3) (y - 3) 7(y - 3) + (y - 3) ( (y - 3)(y + 7) Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial factor. Slide 20 / 128 Example 2 Factor each polynomial. b) Find the GF GF: Reduce each term of the polynomial dividing by the GF In working with common binomial factors, look for factors that are opposites of each other. Slide 21 / 128 For example: (x - y) = - (y - x) because x - y = x + (-y) = -y + x = -(y - x)

8 10 True or False: y - 7 = -1( 7 + y) Slide 22 / 128 True False 11 True or False: 8 - d = -1( d + 8) Slide 23 / 128 True False 12 True or False: 8c - h = -1( -8c + h) Slide 24 / 128 True False

9 13 True or False: -a - b = -1( a + b) Slide 25 / 128 True False Slide 26 / 128 Slide 27 / 128

10 14 If possible, Factor Slide 28 / 128 lready Simplified 15 If possible, Factor Slide 29 / 128 lready Simplified 16 If possible, Factor Slide 30 / 128 lready Simplified

11 17 If possible, Factor Slide 31 / 128 lready Simplified 18 If possible, Factor Slide 32 / 128 lready Simplified 19 If possible, Factor Slide 33 / 128 lready Simplified

12 Slide 34 / 128 Factoring Trinomials: x 2 + bx + c Return to Table of ontents Slide 35 / 128 polynomial that can be simplified to the form ax + bx + c, where a 0, is called a quadratic polynomial. Quadratic term. Linear term. onstant term. 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. Slide 36 / 128 Examples: hoose all of the description that apply. ubic Quadratic Linear onstant Trinomial inomial Monomial

13 20 hoose all of the descriptions that apply to: Slide 37 / 128 E F Quadratic Linear onstant Trinomial inomial Monomial Slide 38 / hoose all of the descriptions that apply to: Slide 39 / 128 E F Quadratic Linear onstant Trinomial inomial Monomial

14 23 hoose all of the descriptions that apply to: Slide 40 / 128 E F Quadratic Linear onstant Trinomial inomial Monomial Simplify. 1) (x + 2)(x + 3) = 2) (x - 4)(x - 1) = 3) (x + 1)(x - 5) = 4) (x + 6)(x - 2) = nswer ank x 2-5x + 4 x 2 + 5x + 6 x 2-4x - 5 x 2 + 4x - 12 Slide 41 / 128 Slide each polynomial from the circle to the correct expression. RELL What did we do?? Look for a pattern!! Slide 42 / 128

15 Slide 43 / 128 Examples: Slide 44 / 128 (x - 8)(x - 1) 24 The factors of 12 will have what kind of signs given the following equation? Slide 45 / 128 oth positive oth Negative igger factor positive, the other negative The bigger factor negative, the other positive

16 25 The factors of 12 will have what kind of signs given the following equation? Slide 46 / 128 oth positive oth negative igger factor positive, the other negative The bigger factor negative, the other positive 26 Factor Slide 47 / 128 (x + 12)(x + 1) (x + 6)(x + 2) (x + 4)(x + 3) (x - 12)(x - 1) E (x - 6)(x - 1) F (x - 4)(x - 3) 27 Factor Slide 48 / 128 (x + 12)(x + 1) (x + 6)(x + 2) (x + 4)(x + 3) (x - 12)(x - 1) E (x - 6)(x - 1) F (x - 4)(x - 3)

17 28 Factor Slide 49 / 128 (x + 12)(x + 1) (x + 6)(x + 2) (x + 4)(x + 3) (x - 12)(x - 1) E (x - 6)(x - 1) F (x - 4)(x - 3) 29 Factor Slide 50 / 128 (x + 12)(x + 1) (x + 6)(x + 2) (x + 4)(x + 3) (x - 12)(x - 1) E (x - 6)(x - 1) F (x - 4)(x - 3) Slide 51 / 128

18 Slide 52 / 128 Examples Slide 53 / The factors of -12 will have what kind of signs given the following equation? Slide 54 / 128 oth positive oth negative igger factor positive, the other negative The bigger factor negative, the other positive

19 31 The factors of -12 will have what kind of signs given the following equation? Slide 55 / 128 oth positive oth negative igger factor positive, the other negative The bigger factor negative, the other positive 32 Factor x 2-4x - 12 Slide 56 / 128 (x + 12)(x - 1) (x + 6)(x - 2) (x + 4)(x - 3) (x - 12)(x + 1) E (x - 6)(x + 2) F (x - 4)(x + 3) 33 Factor Slide 57 / 128 (x + 12)(x - 1) (x + 6)(x - 2) (x + 4)(x - 3) (x - 12)(x + 1) E (x - 6)(x + 1) F (x - 4)(x + 3)

20 34 Factor Slide 58 / 128 (x + 12)(x - 1) (x + 6)(x - 2) (x + 4)(x - 3) (x - 12)(x + 1) E (x - 6)(x + 1) F (x - 4)(x + 3) Slide 59 / 128 Slide 60 / 128 Mixed Practice

21 36 Factor the following Slide 61 / 128 (x - 2)(x - 4) (x + 2)(x + 4) (x - 2)(x +4) (x + 2)(x - 4) 37 Factor the following Slide 62 / 128 (x - 3)(x - 5) (x + 3)(x + 5) (x - 3)(x +5) (x + 3)(x - 5) 38 Factor the following Slide 63 / 128 (x - 3)(x - 4) (x + 3)(x + 4) (x +2)(x +6) (x + 1)(x+12)

22 39 Factor the following Slide 64 / 128 (x - 2)(x - 5) (x + 2)(x + 5) (x - 2)(x +5) (x + 2)(x - 5) Slide 65 / 128 Slide 66 / 128

23 Slide 67 / 128 Factor: Slide 68 / 128 Factor out STEP 1 STEP 2 STEP 3 STEP 4 40 Factor completely: Slide 69 / 128

24 41 Factor completely: Slide 70 / Factor completely: Slide 71 / Factor completely: Slide 72 / 128

25 44 Factor completely: Slide 73 / 128 Slide 74 / 128 Factoring Using Special Patterns Return to Table of ontents When we were multiplying polynomials we had special patterns. Slide 75 / 128 Square of Sums ifference of Sums Product of a Sum and a ifference If we learn to recognize these squares and products we can use them to help us factor.

26 Perfect Square Trinomials The Square of a Sum and the Square of a difference have products that are called Perfect Square Trinomials. Slide 76 / 128 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. Examples of Perfect Square Trinomials Slide 77 / 128 Is the trinomial a perfect square? rag the Perfect Square Trinomials into the ox. Slide 78 / 128 Only Perfect Square Trinomials will remain visible.

27 Slide 79 / Factor Slide 80 / 128 Not a perfect Square Trinomial 46 Factor Slide 81 / 128 Not a perfect Square Trinomial

28 47 Factor Slide 82 / 128 Not a perfect Square Trinomial Slide 83 / 128 Slide 84 / 128 Examples of ifference of Squares

29 Slide 85 / 128 Slide 86 / Factor Slide 87 / 128 Not a ifference of Squares

30 49 Factor Slide 88 / 128 Not a ifference of Squares 50 Factor Slide 89 / 128 Not a ifference of Squares 51 Factor using ifference of Squares: Slide 90 / 128 Not a ifference of Squares

31 Slide 91 / 128 Slide 92 / 128 Factoring Trinomials: ax 2 + bx + c Return to Table of ontents How to factor a trinomial of the form ax² + bx + c. Example: Factor 2d² + 15d + 18 Slide 93 / 128 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, = = = 15 Now substitute into the equation for 15. 2d² + (12 + 3)d + 18 istribute 2d² + 12d + 3d + 18 Group and factor GF 2d(d + 6) + 3(d + 6) Factor common binomial (d + 6)(2d + 3) Remember to check using FOIL!

32 Factor. 15x² - 13x + 2 Slide 94 / 128 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, = = = = -11 Slide 95 / 128 Factor 6y² - 13y - 5 Slide 96 / 128

33 Slide 97 / 128 polynomial that cannot be written as a product of two polynomials is called a prime polynomial. 53 Factor Slide 98 / 128 Prime Polynomial 54 Factor Slide 99 / 128 Prime Polynomial

34 55 Factor Slide 100 / 128 Prime Polynomial Slide 101 / 128 Factoring 4 Term Polynomials Return to Table of ontents Polynomials with four terms like ab - 4b + 6a - 24, can be factored by grouping terms of the polynomials. Slide 102 / 128 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 GF (a - 4) (b + 6) Notice that a - 4 is a common binomial factor and factor!

35 Example 2: 6xy + 8x - 21y - 28 Slide 103 / 128 (6xy + 8x) + (-21y - 28) Group 2x(3y + 4) + (-7)(3y + 4) Factor GF (3y +4) (2x - 7) Factor common binomial 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). Slide 104 / 128 Example 3: 15x - 3xy + 4y - 20 (15x - 3xy) + (4y - 20) Group 3x(5 - y) + 4(y - 5) Factor GF 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. 56 Factor 15ab - 3a + 10b - 2 Slide 105 / 128 (5b - 1)(3a + 2) (5b + 1)(3a + 2) (5b - 1)(3a - 2) (5b + 1)(3a - 1)

36 57 Factor 10m 2 n - 25mn + 6m - 15 Slide 106 / 128 (2m-5)(5mn-3) (2m-5)(5mn+3) (2m+5)(5mn-3) (2m+5)(5mn+3) 58 Factor 20ab - 35b a Slide 107 / 128 (4a - 7)(5b - 9) (4a - 7)(5b + 9) (4a + 7)(5b - 9) (4a + 7)(5b + 9) 59 Factor a 2 - ab + 7b - 7a Slide 108 / 128 (a - b)(a - 7) (a - b)(a + 7) (a + b)(a - 7) (a + b)(a + 7)

37 Slide 109 / 128 Mixed Factoring Return to Table of ontents Summary of Factoring Factor the Polynomial Slide 110 / Terms ifference of Squares Perfect Square Trinomial Factor out GF 3 Terms Factor the Trinomial 4 Terms Group and Factor out GF. Look for a ommon inomial a = 1 a = 1 heck each factor to see if it can be factored again. If a polynomial cannot be factored, then it is called prime. Slide 111 / 128

38 60 Factor completely: Slide 112 / Factor completely Slide 113 / 128 prime polynomial 62 Factor Slide 114 / 128 prime polynomial

39 Slide 115 / Factor Slide 116 / 128 Prime Polynomial Slide 117 / 128 Solving Equations by Factoring Return to Table of ontents

40 Given the following equation, what conclusion(s) can be drawn? ab = 0 Slide 118 / 128 Since the product is 0, one of the factors, a or b, must be 0. This is known as the Zero Product Property. 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 = x = 4 or x = -3 Slide 119 / 128 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 What if you were given the following equation? Slide 120 / 128 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 y the Zero Product Property: x - 6 = 0 or x + 4 = 0 fter solving each equation, we arrive at our solution: {-4, 6}

41 Slide 121 / 128 Slide 122 / 128 Slide 123 / 128

42 65 hoose all of the solutions to: Slide 124 / 128 E F 66 hoose all of the solutions to: Slide 125 / 128 E F 67 hoose all of the solutions to: Slide 126 / 128 E F

43 68 ball is thrown with its height at any time given by Slide 127 / 128 When does the ball hit the ground? -1 seconds 0 seconds 9 seconds 10 seconds Slide 128 / 128