Factors 2 12 Factors Factors 3 13 and Unique Unique 14 4 - PDF document

Slide 1 / 128 Slide 2 / 128 Table of Contents · Factors and GCF · Factoring out GCF's Polynomials · 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 Slide 4 / 128 Number Factors of 15 Factors of 10 Bank 1 11 Factors 2 12 Factors Factors 3 13 and Unique Unique 14 4 to 10 to 15 Greatest Common Factors 15 5 6 16 17 7 18 8 9 19 Factors 10 and 15 10 20 have in common Return to Table of Contents What is the greatest common factor (GCF) of 10 and 15? Slide 5 / 128 Slide 6 / 128 Number Bank Factors of 18 1 What is the GCF of 12 and 15? Factors of 12 1 11 2 12 13 3 Factors Factors 14 Unique 4 Unique to 12 to 18 5 15 6 16 17 7 18 8 9 19 10 20 Factors 12 and 18 have in common What is the greatest common factor (GCF) of 12 and 18?

Slide 7 / 128 Slide 8 / 128 2 What is the GCF of 24 and 48? 3 What is the GCF of 72 and 54? Slide 9 / 128 Slide 10 / 128 4 What is the GCF of 70 and 99? 5 What is the GCF of 28, 56 and 42? Slide 11 / 128 Slide 12 / 128 Variables also have a GCF. 6 What is the GCF of and ? The GCF of variables is the variable(s) that is in each term raised to the lowest exponent given. A B Example: Find the GCF C and and and D and and and

Slide 13 / 128 Slide 14 / 128 Slide 15 / 128 Slide 16 / 128 Factoring out GCFs Return to Table of Contents Slide 17 / 128 Slide 18 / 128 The first step in factoring is to determine its greatest monomial The first step in factoring is to determine its greatest monomial factor. If there is a greatest monomial factor other than 1, use the factor. If there is a greatest monomial factor other than 1, use the distributive property to rewrite the given polynomial as the product distributive property to rewrite the given polynomial as the product of this greatest monomial factor and a polynomial. of this greatest monomial factor and a polynomial. Example 1 Factor each polynomial. Example 1 Factor each polynomial. a) 6x 4 - 15x 3 + 3x 2 4m 3 n - 7m 2 n 2 b) 6x 4 15x 3 3x 2 Find the GCF 3x 2 Find the GCF 3x 2 3x 2 3x 2 GCF: 3x 2 GCF: m 2 n Reduce each term of Reduce each term of the polynomial dividing the polynomial dividing by the GCF by the GCF 3x 2 (2x 2 - 5x + 1) m 2 n(4n - 7n)

Slide 19 / 128 Slide 20 / 128 Sometimes the distributive property can be used to factor a Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has a common binomial polynomial that is not in simplest form but has a common binomial factor. factor. Example 2 Factor each polynomial. Example 2 Factor each polynomial. a) y(y - 3) + 7(y - 3) b) ( Find the GCF Find the GCF ( y(y - 3) 7(y - 3) (y - 3) + GCF: y - 3 GCF: (y - 3) (y - 3) Reduce each term of Reduce each term of the polynomial dividing the polynomial dividing by the GCF by the GCF (y - 3)(y + 7) Slide 21 / 128 Slide 22 / 128 In working with common binomial factors, look for factors that are 10 True or False: y - 7 = -1( 7 + y) opposites of each other. True For example: (x - y) = - (y - x) because x - y = x + (-y) = -y + x = -(y - x) False Slide 23 / 128 Slide 24 / 128 11 True or False: 8 - d = -1( d + 8) 12 True or False: 8c - h = -1( -8c + h) True True False False

Slide 25 / 128 Slide 26 / 128 13 True or False: -a - b = -1( a + b) True False Slide 27 / 128 Slide 28 / 128 14 If possible, Factor A B C D Already Simplified Slide 29 / 128 Slide 30 / 128 15 If possible, Factor 16 If possible, Factor A A B B C C D D Already Simplified Already Simplified

Slide 31 / 128 Slide 32 / 128 17 If possible, Factor 18 If possible, Factor A A B B C C D Already Simplified D Already Simplified Slide 33 / 128 Slide 34 / 128 19 If possible, Factor A B Factoring Trinomials: C x 2 + bx + c D Already Simplified Return to Table of Contents Slide 35 / 128 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. A polynomial that can be simplified to the form ax + bx + c , where a ≠ 0, is called a quadratic polynomial. Cubic Q Linear term. Constant term. u a Quadratic d r a t i c t Linear e r m . Constant Trinomial Binomial Monomial

Slide 37 / 128 Slide 38 / 128 20 Choose all of the descriptions that apply to: Quadratic A Linear B C Constant D Trinomial Binomial E Monomial F Slide 39 / 128 Slide 40 / 128 22 Choose all of the descriptions that apply to: 23 Choose all of the descriptions that apply to: A Quadratic A Quadratic Linear Linear B B Constant Constant C C Trinomial Trinomial D D E Binomial E Binomial F Monomial F Monomial Slide 41 / 128 Slide 42 / 128 Simplify. Answer Bank 1) (x + 2)(x + 3) = _________________________ x 2 - 5x + 4 2) (x - 4)(x - 1) = _________________________ x 2 - 4x - 5 3) (x + 1)(x - 5) = ________________________ x 2 + 5x + 6 x 2 + 4x - 12 4) (x + 6)(x - 2) = ________________________ Slide each polynomial from the circle to the correct expression. RECALL … What did we do?? Look for a pattern!!

Slide 43 / 128 Slide 44 / 128 Examples: (x - 8)(x - 1) Slide 45 / 128 Slide 46 / 128 24 The factors of 12 will have what kind of signs given the 25 The factors of 12 will have what kind of signs given the following equation? following equation? Both positive Both positive A A Both Negative Both negative B B Bigger factor positive, the other negative Bigger factor positive, the other negative C C D The bigger factor negative, the other positive D The bigger factor negative, the other positive Slide 47 / 128 Slide 48 / 128 26 Factor 27 Factor (x + 12)(x + 1) (x + 12)(x + 1) A A (x + 6)(x + 2) (x + 6)(x + 2) B B C (x + 4)(x + 3) C (x + 4)(x + 3) (x - 12)(x - 1) (x - 12)(x - 1) D D (x - 6)(x - 1) (x - 6)(x - 1) E E (x - 4)(x - 3) (x - 4)(x - 3) F F

Slide 49 / 128 Slide 50 / 128 28 Factor 29 Factor (x + 12)(x + 1) (x + 12)(x + 1) A A (x + 6)(x + 2) (x + 6)(x + 2) B B (x + 4)(x + 3) (x + 4)(x + 3) C C D (x - 12)(x - 1) D (x - 12)(x - 1) (x - 6)(x - 1) (x - 6)(x - 1) E E (x - 4)(x - 3) (x - 4)(x - 3) F F Slide 51 / 128 Slide 52 / 128 Slide 53 / 128 Slide 54 / 128 Examples 30 The factors of -12 will have what kind of signs given the following equation? A Both positive Both negative B Bigger factor positive, the other negative C The bigger factor negative, the other positive D

Slide 55 / 128 Slide 56 / 128 Factor x 2 - 4x - 12 31 The factors of -12 will have what kind of signs given the 32 following equation? (x + 12)(x - 1) A Both positive A (x + 6)(x - 2) B B Both negative (x + 4)(x - 3) C Bigger factor positive, the other negative C D (x - 12)(x + 1) The bigger factor negative, the other positive D E (x - 6)(x + 2) (x - 4)(x + 3) F Slide 57 / 128 Slide 58 / 128 33 Factor 34 Factor A (x + 12)(x - 1) A (x + 12)(x - 1) (x + 6)(x - 2) (x + 6)(x - 2) B B (x + 4)(x - 3) (x + 4)(x - 3) C C (x - 12)(x + 1) (x - 12)(x + 1) D D (x - 6)(x + 1) (x - 6)(x + 1) E E F (x - 4)(x + 3) F (x - 4)(x + 3) Slide 59 / 128 Slide 60 / 128 Mixed Practice

Slide 61 / 128 Slide 62 / 128 36 Factor the following 37 Factor the following (x - 2)(x - 4) (x - 3)(x - 5) A A (x + 2)(x + 4) (x + 3)(x + 5) B B (x - 2)(x +4) (x - 3)(x +5) C C D (x + 2)(x - 4) D (x + 3)(x - 5) Slide 63 / 128 Slide 64 / 128 38 Factor the following 39 Factor the following A (x - 3)(x - 4) A (x - 2)(x - 5) (x + 3)(x + 4) (x + 2)(x + 5) B B (x +2)(x +6) (x - 2)(x +5) C C (x + 1)(x+12) (x + 2)(x - 5) D D Slide 65 / 128 Slide 66 / 128

Slide 67 / 128 Slide 68 / 128 Factor: Factor out STEP STEP STEP STEP 3 4 2 1 Slide 69 / 128 Slide 70 / 128 40 Factor completely: 41 Factor completely: A A B B C C D D Slide 71 / 128 Slide 72 / 128 42 Factor completely: 43 Factor completely: A A B B C C D D

Slide 73 / 128 Slide 74 / 128 44 Factor completely: A B Factoring Using C Special Patterns D Return to Table of Contents Slide 75 / 128 Slide 76 / 128 Perfect Square Trinomials When we were multiplying polynomials we had The Square of a Sum and the Square of a difference have special patterns. products that are called Perfect Square Trinomials. How to Recognize a Perfect Square Trinomial: Square of Sums Recall: 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. 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 Slide 78 / 128 Is the trinomial a perfect square? Examples of Perfect Square Trinomials Drag the Perfect Square Trinomials into the Box. Only Perfect Square Trinomials will remain visible.

Slide 79 / 128 Slide 80 / 128 45 Factor A B C Not a perfect Square Trinomial D Slide 81 / 128 Slide 82 / 128 46 Factor 47 Factor A A B B C C Not a perfect Square Trinomial Not a perfect Square Trinomial D D Slide 83 / 128 Slide 84 / 128 Examples of Difference of Squares