Algebra I. Slide 1 / 211. Slide 2 / 211. Slide 3 / 211. Polynomials. Table of Contents. New Jersey Center for Teaching and Learning

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1 New Jersey enter for Teaching and Learning Slide 1 / 211 Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. lick to go to website: Slide 2 / 211 lgebra I Polynomials Table of ontents efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials Mulitplying a Polynomial by a Monomial Multiplying Polynomials Special inomial Products Solving Equations Factors and GF Factoring out GF's Identifying & Factoring x2+ bx + c Factoring Using Special Patterns Factoring Trinomials ax2 + bx + c Factoring 4 Term Polynomials Mixed Factoring Solving Equations by Factoring Slide 3 / 211

2 Slide 4 / 211 efinitions of Monomials, Polynomials and egrees Return to Table of ontents Slide 5 / 211 monomialis a one- term ex pression form ed by a num ber, a variable, or the product of num bers and variables. Ex am ples of m onom ials... 81y 4 z 2 1 7x 32 rt x,4 5 7 mn 3 rag the following term s into the correct sorting box. If you sort correctly, the term will be visible. If you sort incorrectly, the term will ) t + 7y 7 x 3y 5-12 x2 (5 disappear. - 4 a+ b -5 5x rs x y Monom ials x yz 2 c 2) 4 (5 a b Slide 6 / 211

3 Slide 7 / 211 polynomial is an expression that contains two or more monomials. d + c Examples of polynomials a2 2 x 3+ x 8 7+ b+ c d3 a4b rt a 3-2b2 4 c- m n 3 Slide 8 / 211 egrees 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? 1 The variable x has a degree 1. 2) The degree of -6x3y is? 3) The degree of 9 is? 4 The x has a power of 3 and the y has a power of 1, so the degree is 3+1 =4. 0 constant has a degree 0, because there is no variable. Slide 9 / What is the degree of ?

4 Slide 10 / What is the degree of ? Slide 11 / What is the degree of 3? Slide 12 / What is the degree of?

5 Slide 13 / 211 egrees 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 3y2-6xy2 + xy. The monomial 4x3y2 has a degree of 5, the monomial 6xy2 has a degree of 3, and the monomial xy has a degree of 2. The highest degree is 5, so the degree of the polynomial is 5. Slide 14 / 211 Find the degree of each polynomial nswers: 1) 0 1) 3 3 2) 3 2) 12c 3) ab 3) 2 4) 8s4 t 4) 5 5) 2-7n 5) 1 6) h - 8t 6) 4 7) s3 + 2v2 y2-1 7) 4 4 Slide 15 / What is the degree of the following polynomial:

6 Slide 16 / What is the degree of the following polynomial: Slide 17 / What is the degree of the following polynomial: Slide 18 / What is the degree of the following polynomial:

7 Slide 19 / 211 dding and Subtracting Polynomials Return to Table of ontents Slide 20 / 211 Standard Form The standard form of an equation is to put the terms in order from highest degree to the lowest degree. Standard form is commonly excepted way to write polynomials. Example: is in standard form. Put the following equation into standard form: Slide 21 / 211 Monomials with the same variables and the same power are like terms. Like TermsUnlike Terms 4x and -12x x3y and 4x3y -3b and 3a 6a2b and -2ab2

8 Slide 22 / 211 ombine these like terms using the indicated operation. Slide 23 / Simplify Slide 24 / Simplify

9 Slide 25 / Simplify Slide 26 / 211 To add polynomials, combine the like terms from each polynomial. To add vertically, first line up the like terms and then add. Examples: (3x2 +5x -12) + (5x 2-7x +3) line up the like terms 3x2 + 5x - 12 (+) 5x2-7x + 3 8x2-2x - 9 (3x 4-5x) + (7x 4 +5x2-14x) line up the like terms 4 3x -5x (+) 7x4 +5x2-14x 4 10x +5x2-19x = Slide 27 / 211 We can also add polynomials horizontally. (3x2 + 12x - 5) + (5x2-7x - 9) Use the communitive and associative properties to group like terms. (3x2 + 5x2) + (12x + -7x) + ( ) 8x2 + 5x - 14

10 Slide 28 / dd Slide 29 / dd Slide 30 / dd

11 Slide 31 / dd Slide 32 / dd Slide 33 / 211 To subtract polynomials, subtract the coefficients of like terms. Example: -3x - 4x = -7x 13y - (-9y) = 22y 6xy - 13xy = -7xy

12 We can subtract polynomials vertically and horizontally. Slide 34 / 211 To subtract a polynomial, change the subtraction to adding -1. istribute the -1 and then follow the rules for adding polynomials (3x2 +4x -5) - (5x 2-6x +3) (3x2+4x-5) +(-1) (5x 2-6x+3) (3x2+4x-5) + (-5x 2+6x-3) 3x 2 + 4x - 5 (+) -5x2-6x + 3-2x 2 +10x - 8 We can subtract polynomials vertically and horizontally. Slide 35 / 211 To subtract a polynomial, change the subtraction to adding -1. istribute the -1 and then follow the rules for adding polynomials (4x3-3x -5) - (2x 3 +4x 2-7) (4x3-3x -5) +(-1)(2x 3 +4x 2-7) (4x3-3x -5) + (-2x 3-4x 2 +7) 4x 3-3x - 5 (+) -2x3-4x x 3-4x 2-3x + 2 Slide 36 / 211 We can also subtract polynomials horizontally. (3x2 + 12x - 5) - (5x 2-7x - 9) hange the subtraction to adding a negative one and distribute the negative one. (3x2 + 12x - 5) +(-1)(5x 2-7x - 9) (3x2 + 12x - 5) + (-5x 2 + 7x + 9) Use the communitive and associative properties to group like terms. (3x2 +-5x2) + (12x +7x) + (-5 +9) -2x2 + 19x + 4

13 Slide 37 / Subtract Slide 38 / Subtract Slide 39 / Subtract

14 Slide 40 / Subtract Slide 41 / Subtract Slide 42 / What is the perimeter of the following figure? (answers are in units)

15 Slide 43 / 211 Multiplying a Polynomial by a Monomial Return to Table of ontents Slide 44 / 211 Find the total area of the rectangles square units square units Slide 45 / 211 To m ultiply a polynom ial by a m onom ial, you use the distributive property together with the laws of ex ponents for m ultiplication. Examples: Simplify. - 2x(5x2-6x + 8) - 2x(5x x + 8) (- 2x)(5x2 ) + -( 2x)(- 6x ) + - (2x)(8) - 10x3 + 12x x - 10x3 + 12x2-16x

16 Slide 46 / 211 To m ultiply a polynom ial by a m onom ial, you use the distributive property together with the laws of ex ponents for m ultiplication. Examples: Simplify. - 3x2 (- 2x2 + 3x - 12) - 3x2 (- 2x2 + 3x ) (- 3x2 )(- 2x2 ) + -( 3x2)(3x ) + -( 3x2)(- 12) 3 6x x + 36x2 6x4-9x3 + 36x2 Slide 47 / 211 Let's Try It! Multiply to sim plify x 4 + 4x 3-7x x2 (5x2-6x - 3) Slide check. 3 20x 4 - to 24x - 12x Slide to check. 3. 3x y(4x3 y2-5x2 y3 + 8x 4y) Slide to 3check x y - 15x y x2 y 5 Slide 48 / What is the area of the rectangle shown? x2 + 2 x + 4 x2

17 Slide 49 / x 2 + 8x x 2 + 8x x 2 + 8x 2-12x 6x 3 + 8x 2-12x Slide 50 / Slide 51 /

18 Slide 52 / Find the area of a triangle (=1 /2 bh) with a base of 4x and a height of 2x - 8. ll answers are in square units. Slide 53 / 211 Multiplying Polynomials Return to Table of ontents Find the total area of the rectangles (2 + 6) (5 + 8 ) = 2 (5 + 8) + 6 (5 + 8) = 2(5) + 2(8) + 6(5) + 6(8) = = 148 sq.units rea of the big rectangle rea of the horizontal rectangles rea of each box Slide 54 / 211

19 Let us observe the work from the previous example, Slide 55 / 211 (2 + 6) (5 + 8 ) = 2 (5 + 8) + 6 (5 + 8) = 2(5) + 2(8) + 6(5) + 6(8) = = 148 sq.units From to, we changed the problem so that instead of a polynomial times a polynomial, we now have a monomial times a polynomial. Use this to help solve the next example. Slide 56 / 211 Find the total area of the rectangles. 2x 4 x 3 To m ultiply a polynom ial by a polynom ial, you m ultiply each term of the first polynom ials by each term of the second. Then, add like term s. Example 1 : (2x + 4y)(3x + 2y) 2x(3x + 2y) + 4y(3x + 2y) 2x(3x) + 2x(2y) + 4y(3x) + 4y(2y) 6x 2 + 4xy + 12xy + 8y 2 6x xy + 8y 2 Example 2 : Slide 57 / 211

20 The FOIL Method can be used to remember how multiply two binomials. To multiply two binomials, find the sum of... First terms Outer terms Example: First Outer Inner Inner Terms Slide 58 / 211 Last Terms Last Slide 59 / 211 Try it!find each product. 1) (x - 4)(x - 3) 2-3x - 8) 2) (x + 2)(2x x2 Slide - 7x +to12 check. Slide to check. 2x3 + x2-14x - 16 Try it!find each product. 3) (2x - 3y)(4x + 5y) 2-2x y - 15y 2 8x Slide to check. 2-2x + 4) 4) (x2 + 3x - 6)(x x4 + x3-8x2 + 24x - 24 Slide to check. Slide 60 / 211

21 Slide 61 / What is the total area of the rectangles shown? 4x 5 2x 4 Slide 62 / Slide 63 /

22 Slide 64 / Slide 65 / Slide 66 / Find the area of a square with a side of

23 Slide 67 / What is the area of the rectangle (in square units)? 3x2 + 5x + 2 3x2 + 6x + 2 3x2-6x + 2 3x2-5x +2 Slide 68 / 211 How would we find the area of the shaded region? Shaded rea = Total area - Unshaded rea sq. units Slide 69 / What is the area of the shaded region (in sq. units)? 11x 2 + 3x - 8 7x 2 + 3x - 9 7x 2-3x x 2-3x - 8

24 Slide 70 / What is the area of the shaded region (in square units)? 2x 2-2x - 8 2x 2-4x - 6 2x 2-10x - 8 2x 2-6x - 4 Slide 71 / 211 Special inomial Products Return to Table of ontents Slide 72 / 211 Square of a Sum (a + b)2 (a + b)(a + b) a2 + 2ab + 2b The square of a + b is the square of a plus twice the product of a and b plus the square of b. 2 Ex am ple: (5x + 3) (5x + 3)(5x + 3) 25x2 + 30x + 9

25 Slide 73 / 211 Square of a ifference (a - b)2 (a - b)(a - b) a2-2ab + 2b The square of a - b is the square of a m inus twice the product of a and b plus the square of b. 2 (7x - 4) (7x - 4)(7x - 4) 49x2-56x + 16 Ex am ple: Slide 74 / 211 Product of a Sum and a ifference (a + b)(a - b) a2 + - ab + ab +2 - bnotice the - aband ab a2 - b2 equals 0. The product of a + b and a - b is the square of a m inus the square of b. Ex am ple: (3y - 8)(3y + 8) Rem em ber the inner and 9y2-64outer term s equals 0. Slide 75 / 211 Try It! Find each product. 1. (3p + 9)2 2 9pSlide + 54p + 81 to check (6 - p) 36Slide - 12p +2p to check. 3. (2x - 3)(2x + 3) Slide 4x2 to- check. 9

26 Slide 76 / x x + 10x x 2-10x + 25 x 2-25 Slide 77 / Slide 78 / What is the area of a square with sides 2x + 4?

27 Slide 79 / Slide 80 / 211 Solving Equations Return to Table of ontents Slide 81 / 211 Given the following equation, what conclusion(s) can be drawn? ab = 0 Since the product is 0, one of the factors, a or b, m ust be 0. This is known as the Zero Product Property.

28 Slide 82 / 211 Zero Product Property Rule: If ab=0, then either a=0 or b=0 Given the following equation, what conclusion(s) can be drawn? (x - 4)(x + 3) = 0 Slide 83 / 211 Since the product is 0, one of the factors m ust be 0. Therefore, either x - 4 = or 0 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 (x -the 4)(xoriginal + 3) = equation. 0 To check x = 4: (x - 4)(x + 3) = 0 solution into To check xback = - 3: (4-4)(4 + 3) = 0 (0)(7) = 0 0= 0 (- 3-4)( ) = 0 (- 7)(0) = 0 0= 0 What if you were given the following equation? (x - 6)(x + 4) = 0 y the Zero Product Property: x - 6= 0 x = 6 x = -4 or x + 4= 0 fter solving each equation, we arrive at our solution: {- 4, 6} Slide 84 / 211

29 Slide 85 / Solve (a + 3)(a - 6) = 0. {3, 6} {-3, -6} {-3, 6} {3, -6} Slide 86 / Solve (a - 2)(a - 4) = 0. {2, 4} {-2, -4} {-2, 4} {2, -4} Slide 87 / Solve (2a - 8)(a + 1) = 0. {-1, -16} {-1, 16} {-1, 4} {-1, -4}

30 Slide 88 / 211 Factors and Greatest ommon Factors Return to Table of ontents Factors of 10 Factors of 15 Factors Unique to 15 Factors Unique to 10 Factors 10 and 15 have in common Number ank Slide 89 / What is the greatest common factor (GF) of 10 and 15? Factors of 12 Factors Unique to 12 Factors of 18 Factors Unique to 18 Factors 12 and 18 have in common Number ank What is the greatest common factor (GF) of 12 and 18? Slide 90 / 211

31 Slide 91 / What is the GF of 12 and 15? Slide 92 / What is the GF of 24 and 48? Slide 93 / What is the GF of 72 and 54?

32 Slide 94 / What is the GF of 70 and 99? Slide 95 / What is the GF of 28, 56 and 42? Slide 96 / 211 Variables also have a GF. 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

33 49 What is the GF of and Slide 97 / 211? 50 What is the GF of and Slide 98 / 211? 51 What is the GF of and and Slide 99 / 211?

34 52 What is the GF of and and Slide 100 / 211? Slide 101 / 211 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. Example 1 Factor each polynomial. a) 6 x4-1 5 x3 + 3 x2 Find the GF GF: 3x 3x 2 2 6x 4 15x 3 3x 2 3x 2 3x 2 3x 2 Reduce each term of the polynomial dividing by the GF 3 x 2 (2 x 2-5x + 1) Slide 102 / 211

35 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 103 / 211 Example 1 Factor each polynomial b) 4 m n - 7 m n Find the GF GF: m2n Reduce each term of the polynomial dividing by the GF m2 n(4 n - 7 n) Slide 104 / 211 Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has common a binomial factor. Example 2 Factor each polynom ial. y(y - 3) + 7(y - 3) Find the GF (y - 3) GF: y - 3 ( y(y - 3) (y - 3) + 7(y - 3) (y - 3) ( a) Reduce each term of the polynomial dividing by the GF (y - 3)(y + 7) Slide 105 / 211 Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has common a binomial factor. Example 2 Factor each polynom ial. b) Find the GF GF: Reduce each term of the polynomial dividing by the GF

36 In working with common binomial factors, look for factors that are opposites of each other. Slide 106 / 211 For example: (x - y) = - (y - x ) because x - y = x + (- y) = - y + x = - (y - x) Slide 107 / True or False: y - 7 = -1( 7 + y) True False Slide 108 / True or False: 8 - d = -1( d + 8) True False

37 Slide 109 / True or False: 8c - h = -1( -8c + h) True False Slide 110 / True or False: -a - b = -1( a + b) True False 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) Example 3 Factor each polynomial. a) n(n - 3) - 7(3 - n) Find the GF GF: Reduce each term of the polynomial dividing by the GF (n - 3)(n + 7) Slide 111 / 211

38 Slide 112 / 211 In working with common binomial factors, look for factors that are opposites of each other. For example: (x - y) = - (y - because x) x - y = x + (- y) = - y + x = - (y - x) Example 3 Factor each polynomial. b) p(h - 1) + 4(1 - h) Find the GF GF: Reduce each term of the polynomial dividing by the GF (h - 1)(p - 4) Slide 113 / If possible, Factor lready Simplified Slide 114 / If possible, Factor lready Simplified

39 Slide 115 / If possible, Factor lready Simplified Slide 116 / If possible, Factor lready Simplified Slide 117 / If possible, Factor lready Simplified

40 Slide 118 / 211 Identifying & Factoring: x2 + bx + c Return to Table of ontents Slide 119 / 211 polynom ial that can be sim plified to the form ax + bx + c, where a # 0, is called quadratic a polynom.ial Li n Qu ea o n s ad rt ra er t an t t ic m. te te r m. rm. quadratic polynom ial in which b # 0 and c # 0 is called a quadratic trinom.ial If only b= 0 or c= 0 it is called quadratic a binom ial. If both b= 0 and c= 0 it is quadratic a m onom.ial Ex am ples: hoose all of the description that apply. ubic Quadratic Linear onstant Trinomial inomial Monomial Slide 120 / 211

41 Slide 121 / hoose all of the descriptions that apply to: Quadratic Linear onstant Trinomial E inomial F Monomial Slide 122 / hoose all of the descriptions that apply to: Quadratic Linear onstant Trinomial E inomial F Monomial Slide 123 / hoose all of the descriptions that apply to: Quadratic Linear onstant Trinomial E inomial F Monomial

42 Slide 124 / hoose all of the descriptions that apply to: Quadratic Linear onstant Trinomial E inomial F Monomial Simplify. nswer ank 1 ) (x + 2 )(x + 3 ) = 2 ) (x - 4 )(x - 1 ) = Slide 125 / 211 x2-5 x + 4 x2-4 x ) (x + 1 )(x - 5 ) = x2 + 5 x ) (x + 6 )(x - 2 ) = x2 + 4 x Slide each polynomial from the circle to the correct expression. RELL What did we do?? Look for a pattern!! To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: Factors of 6 have the same signs. Factors of 6 Sum to 5? 1, 6 7 2, 3 5 Factors of 6 add to +5. oth factors must be positive. Slide 126 / 211

43 Slide 127 / 211 To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: Factors of 6 have the same signs. Factors of 6 Sum to -7? -1, , -3-5 Factors of 6 add to -7. oth factors must be negative. Slide 128 / 211 Examples: (x - 8)(x - 1) Slide 129 / The factors of 12 will have what kind of signs given the following equation? oth positive oth Negative igger factor positive, the other negative The bigger factor negative, the other positive

44 Slide 130 / The factors of 12 will have what kind of signs given the following equation? oth positive oth negative igger factor positive, the other negative The bigger factor negative, the other positive Slide 131 / Factor (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 132 / Factor (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)

45 Slide 133 / Factor (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 134 / Factor (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) To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: Factors of 6 have the opposite signs. Factors of 6 Sum to -5? 1, , -3-1 Factors of 6 add to -5. Larger factor must be negative. Slide 135 / 211

46 Slide 136 / 211 To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: Factors of 6 have the opposite signs. Factors of 6 Sum to 1? -1, 6 5-2, 3 1 Factors of 6 add to +1. Larger factor must be positive. Slide 137 / 211 Examples Slide 138 / The factors of -12 will have what kind of signs given the following equation? oth positive oth negative igger factor positive, the other negative The bigger factor negative, the other positive

47 Slide 139 / The factors of -12 will have what kind of signs given the following equation? oth positive oth negative igger factor positive, the other negative The bigger factor negative, the other positive Slide 140 / Factor (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 141 / Factor (x + 12)(x - 1) (x + 6)(x - 2) (x + 4)(x - 3) (x - 12)(x + 1) E (x - 6)(x + 1) F unable to factor using this method

48 Slide 142 / Factor (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 143 / 211 Mixed Practice Slide 144 / Factor the following (x - 2)(x - 4) (x + 2)(x + 4) (x - 2)(x +4) (x + 2)(x - 4)

49 Slide 145 / Factor the following (x - 3)(x - 5) (x + 3)(x + 5) (x - 3)(x +5) (x + 3)(x - 5) Slide 146 / Factor the following (x - 3)(x - 4) (x + 3)(x + 4) (x +2)(x +6) (x + 1)(x+12) Slide 147 / Factor the following (x - 2)(x - 5) (x + 2)(x + 5) (x - 2)(x +5) (x + 2)(x - 5)

50 Slide 148 / 211 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. There is no common STEP monomial,so factor: 1 STEP 2 STEP 3 STEP 4 Slide 149 / 211 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. There is no common STEP monomial,so factor:1 STEP 2 STEP 3 STEP 4 Slide 150 / 211 Factor: Factor out STEP 1 STEP 2 STEP 3 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. STEP 4

51 Slide 151 / 211 Factor: Factor out STEP 1 STEP 2 STEP 3 STEP 4 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. Slide 152 / Factor completely: Slide 153 / Factor completely:

52 Slide 154 / Factor completely: Slide 155 / Factor completely: Slide 156 / Factor completely:

53 Slide 157 / 211 Factoring Using Special Patterns Return to Table of ontents Slide 158 / 211 When we were multiplying polynomials we had special patterns. 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. 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 159 / 211

54 Slide 160 / 211 Examples of Perfect Square Trinomials Is the trinomial a perfect square? Slide 161 / 211 rag the Perfect Square Trinomials into the ox. Only Perfect Square Trinomials will remain visible. Factoring Perfect Square Trinomials. Once a Perfect Square Trinomial has been identified, it factors following the form: (sq rt of the first term sign of the middle term sq rt of the third term)2 Examples: Slide 162 / 211

55 Slide 163 / Factor Not a perfect Square Trinomial Slide 164 / Factor Not a perfect Square Trinomial Slide 165 / Factor Not a perfect Square Trinomial

56 ifference of Squares Slide 166 / 211 The Product of a Sum and a ifference is a difference of Squares. ifference of Squares is recognizable by seeing each term in the binomial are perfect squares and the operation is subtraction. Slide 167 / 211 Examples of ifference of Squares Is the binomial a difference of squares? rag the ifference of Squares binomials into the ox. Only ifference of Squares will remain visible. Slide 168 / 211

57 Slide 169 / 211 Factoring a ifference of Squares Once a binomial is determined to be a ifference of Squares, it factors following the pattern: (sq rt of 1st term - sq rt of 2nd term)(sq rt of 1st term + sq rt of 2nd term) Examples: Slide 170 / Factor Not a ifference of Squares Slide 171 / Factor Not a ifference of Squares

58 Slide 172 / Factor Not a ifference of Squares Slide 173 / Factor using ifference of Squares: Not a ifference of Squares Slide 174 / Factor

59 Slide 175 / 211 Factoring Trinomials: ax2 + bx + c Return to Table of ontents How to factor a trinomial of the form ax² + bx + c. Example: Slide 176 / 211 Factor 2d² + 15d = 36 Now find two integers whose product is 36 and whose sum is equal to b or 15. Factors of 3 6 Sum = 1 5? = 37 1, = 20 2, = 15 3, 12 Now substitute into the equation for d² + ( )d istribute 2 d² d + 3 d Group and factor GF2 d(d + 6 ) + 3 (d + 6 ) Factor common binomial (d + 6 )(2 d + 3 ) Remember to check using FOIL! Slide 177 / 211 Factor. 1 5 x² x + 2 a = 1 5 and c = 2, but b = Since both a and c are positive, and b is negative we need to find two negative factors of 3 0 that add up to Sum = - 1 3? Factors of 3 0-1, 2, 3, 5, = 15 = 10 = 6 =

60 Factor. 2 b - b Slide 178 / 211 a = 2, c = - 1 0, and b = - 1 Since a times c is negative, and b is negative we need to find two factors with opposite signs whose product is and that add up to - 1. Since the sum is negative, larger factor of must be negative. Factors of Sum = - 1? 1, , , = = = -1 Slide 179 / 211 Factor 6 y² y - 5 Slide 180 / 211 polynomial that cannot be written as a product of two polynomials is called aprime polynomial.

61 Slide 181 / Factor Prime Polynomial Slide 182 / Factor Prime Polynomial Slide 183 / Factor Prime Polynomial

62 Slide 184 / 211 Factoring 4 Term Polynomials Return to Table of ontents Polynom ials with four term s likeab - 4b + 6a - 24, can be factored by grouping term s of the polynom ials. Slide 185 / 211 Example 1 : ab - 4 b + 6 a (ab - 4b) + (6a - 24) Group term s into binom ials that can be factored using the distributive property b(a - 4) + 6(a - 4) Factor the GF (a - 4) (b + 6) Notice that a - 4 is a com m on binom ial factor and factor! Example 2 : 6x y + 8x - 21y - 28 (6x y + 8x ) + (- 21y - Group 28) 2x (3y + 4) + (- 7)(3y +Factor 4) GF (3y + 4) (2x - 7)Factor com m on binom ial Slide 186 / 211

63 Slide 187 / 211 You m ust be able to recognize additive inverses!!! (3 - aand a - 3are additive inverses because their sum is equal to zero.) Rem em ber3 - a = - 1(a -. 3) Example 3 : 15x - 3x y + 4y - 20 (15x - 3x y) + (4y - 20) Group 3x (5 - y) + 4(y - Factor 5) GF 3x (- 1)(- 5 + y) + 4(y -Notice 5) additive inverses - 3x (y - 5) + 4(y -Sim 5) plify (y - 5) (- 3x + 4)Factor com m on binom ial Remember to check each problem by using FOIL. Slide 188 / Factor 15ab - 3a + 10b - 2 (5b - 1)(3a + 2) (5b + 1)(3a + 2) (5b - 1)(3a - 2) (5b + 1)(3a - 1) Slide 189 / Factor 10m2n - 25mn + 6m - 15 (2m-5)(5mn-3) (2m-5)(5mn+3) (2m+5)(5mn-3) (2m+5)(5mn+3)

64 Slide 190 / Factor 20ab - 35b a (4a - 7)(5b - 9) (4a - 7)(5b + 9) (4a + 7)(5b - 9) (4a + 7)(5b + 9) Slide 191 / Factor a2 - ab + 7b - 7a (a - b)(a - 7) (a - b)(a + 7) (a + b)(a - 7) (a + b)(a + 7) Slide 192 / 211 Mixed Factoring Return to Table of ontents

65 Summary of Factoring Slide 193 / 211 Factor the Polynomial Factor out GF 4 Terms 2 Terms ifference of Squares 3 Terms Perfect Square Trinomial Factor the Trinomial a=1 Group and Factor out GF. Look for a ommon inomial 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 194 / 211 Examples 3r 3-9r 2 + 6r 3r(r 2-3r + 2) 3r(r - 1)(r - 2) Slide 195 / Factor completely:

66 Slide 196 / Factor completely prime polynomial Slide 197 / Factor prime polynomial Slide 198 / Factor completely 10w 2(x2-10x +100)

67 Slide 199 / Factor Prime Polynomial Slide 200 / 211 Solving Equations by Factoring Return to Table of ontents Slide 201 / 211 Given the following equation, what conclusion(s) can be drawn? ab = 0 Since the product is 0, one of the factors, a or b, m ust be 0. This is known as the Zero Product Property.

68 Recall ~ Given the following equation, what conclusion(s) can be drawn? (x - 4)(x + 3) = 0 Slide 202 / 211 Since the product is 0, one of the factors m ust be 0. Therefore, either x - 4= 0 or x + 3 =.0 x - 4 = 0 or x = 4 or x + 3 = 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 To check x = 4: (x - 4)(x + 3) = 0 (4-4)(4 + 3) = 0 (0)(7) = 0 0= 0 (- 3-4)( ) = 0 (- 7)(0) = 0 0= 0 What if you were given the following equation? Slide 203 / 211 How would you solve it? We can use the Zero Product Property to solve it. How can we turn this polynom ial into a m ultiplication problem? Factor it! 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} Slide 204 / 211 Solve 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.

69 Slide 205 / 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 Slide 206 / hoose all of the solutions to: E F Slide 207 / hoose all of the solutions to: E F

70 Slide 208 / hoose all of the solutions to: E F pplication~ 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? Slide 209 / 211 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. Slide 210 / ball is thrown with its height at any time given by When does the ball hit the ground? -1 seconds 0 seconds 9 seconds 10 seconds

71 Slide 211 / 211