Slide 1 / 11 Slide / 11 New Jersey enter for Teaching and Learning lgebra I Progressive Mathematics Initiative Polynomials This material is made freely available at www.njctl.org 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. 01-07-1 www.njctl.org lick to go to website: www.njctl.org Slide / 11 Slide 4 / 11 Table of ontents efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials efinitions of Monomials, Polynomials and egrees Mulitplying a Polynomial by a Monomial Multiplying Polynomials Special inomial Products Solving Equations Factors and GF Factoring out GF's Return to Table of ontents Identifying & Factoring x+ bx + c Factoring Using Special Patterns Factoring Trinomials ax + bx + c Factoring 4 Term Polynomials Mixed Factoring Solving Equations by Factoring Slide 5 / 11 Slide 6 / 11 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 y 5-1 x (5 disappear. - 4 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 1 7x rt 6 8 4x,4 5 a+ b -5 5 7 mn x + 7 6+ 5 16 rs x y4 15 7 Monom ials 4 8 x yz c ) 4 (5 a b
Slide 7 / 11 Slide 8 / 11 polynomial is an expression that contains two or more monomials. d c + Examples of polynomials.... 5+ a 8x x + 7+ b+ c+ 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 1 is 0. The constant 0 has no degree. 4 d Examples: 8a - b a4b rt + 15 6 4 c- m n 1) The degree of x is? 1 The variable x has a degree 1. ) The degree of -6xy is? ) The degree of 9 is? Slide 9 / 11 1 What is the degree of 0 1? What is the degree of 0 1 0 1? Slide 1 / 11 4 What is the degree of? 0 constant has a degree 0, because there is no variable. Slide 10 / 11 Slide 11 / 11 4 The x has a power of and the y has a power of 1, so the degree is +1 =4. What is the degree of?
Slide 1 / 11 Slide 14 / 11 Find the degree of each polynomial nswers: 1) 0 1) ) ) 1c egrees of Polynomials The degree of a polynomial is the same as that of the term with the greatest degree. ) ab ) 4) 8s t 4) 5 5) - 7n 5) 1 6) h4-8t 6) 4 7) s + v y - 1 7) 4 4 Example: Find degree of the polynomial 4x y - 6xy + xy. The monomial 4xy has a degree of 5, the monomial 6xy has a degree of, and the monomial xy has a degree of. The highest degree is 5, so the degree of the polynomial is 5. Slide 15 / 11 5 Slide 16 / 11 What is the degree of the following polynomial: 4 6 What is the degree of the following polynomial: 4 5 5 6 6 Slide 17 / 11 7 Slide 18 / 11 What is the degree of the following polynomial: 8 What is the degree of the following polynomial: 4 4 5 5 6 6
Slide 19 / 11 Slide 0 / 11 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. dding and Subtracting Polynomials Example: is in standard form. Put the following equation into standard form: Return to Table of ontents Slide 1 / 11 Slide / 11 Monomials with the same variables and the same power are like terms. Like TermsUnlike Terms 4x and -1x xy and 4xy ombine these like terms using the indicated operation. -b and a 6ab and -ab Slide / 11 9 Slide 4 / 11 Simplify 10 Simplify
Slide 5 / 11 11 Slide 6 / 11 Simplify To add polynomials, combine the like terms from each polynomial. To add vertically, first line up the like terms and then add. Examples: (x +5x -1) + (5x -7x +) line up the like terms x + 5x - 1 (+) 5x - 7x + 8x - x - 9 (x 4-5x) + (7x 4 +5x -14x) line up the like terms 4 x -5x (+) 7x4 +5x - 14x 4 10x +5x - 19x = Slide 7 / 11 Slide 8 / 11 1 We can also add polynomials horizontally. (x + 1x - 5) + (5x - 7x - 9) dd Use the communitive and associative properties to group like terms. (x + 5x) + (1x + -7x) + (-5 + -9) 8x + 5x - 14 Slide 9 / 11 1 Slide 0 / 11 dd 14 dd
Slide 1 / 11 15 Slide / 11 dd 16 dd Slide / 11 To subtract polynomials, subtract the coefficients of like terms. Example: -x - 4x = -7x 1y - (-9y) = y 6xy - 1xy = -7xy Slide 4 / 11 We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to adding -1. istribute the -1 and then follow the rules for adding polynomials (x +4x -5) - (5x -6x +) (x+4x-5) +(-1) (5x -6x+) (x+4x-5) + (-5x +6x-) x + 4x - 5 (+) -5x - 6x + -x +10x - 8 Slide 5 / 11 We can subtract polynomials vertically and horizontally. To subtract a polynomial, change the subtraction to adding -1. istribute the -1 and then follow the rules for adding polynomials (4x -x -5) - (x +4x -7) (4x -x -5) +(-1)(x +4x -7) (4x -x -5) + (-x -4x +7) 4x - x - 5 (+) -x - 4x +7 x - 4x - x + Slide 6 / 11 We can also subtract polynomials horizontally. (x + 1x - 5) - (5x - 7x - 9) hange the subtraction to adding a negative one and distribute the negative one. (x + 1x - 5) +(-1)(5x - 7x - 9) (x + 1x - 5) + (-5x + 7x + 9) Use the communitive and associative properties to group like terms. (x +-5x) + (1x +7x) + (-5 +9) -x + 19x + 4
Slide 7 / 11 17 Slide 8 / 11 Subtract 18 Subtract Slide 9 / 11 19 Slide 40 / 11 Subtract 0 Subtract Slide 4 / 11 Subtract Slide 41 / 11 1 What is the perimeter of the following figure? (answers are in units)
Slide 4 / 11 Slide 44 / 11 Find the total area of the rectangles. Multiplying a Polynomial by a Monomial 5 4 8 square units Return to Table of ontents square units Slide 45 / 11 Slide 46 / 11 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. 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: Examples: Simplify. Simplify. - x(5x - 6x + 8) - x (- x + x - 1) - x(5x + - 6x + 8) - x (- x + x + - 1) (- x)(5x ) + -( x)(- 6x ) + - (x)(8) (- x )(- x ) + -( x)(x ) + -( x)(- 1) - 10x + 1x + - 16x 6x4 + - 9x + 6x - 10x + 1x - 16x 6x4-9x + 6x Slide 47 / 11 Let's Try It! Multiply to sim plify. 1. - x 4 + 4x - 7x Slide to check. Slide 48 / 11 What is the area of the rectangle shown?. 4x (5x - 6x - ). x y(4x y - 5x y + 8x 4y) Slide check. 0x 4 - to 4x - 1x Slide to check. 4 1x y - 15x y 4 + 4x y 5 x + x + 4 x
Slide 49 / 11 Slide 50 / 11 4 5 6x + 8x - 1 6x + 8x - 1 6x + 8x - 1x 6x + 8x - 1x Slide 51 / 11 Slide 5 / 11 6 7Find the area of a triangle (=1 / bh) with a base of 4x and a height of x - 8. ll answers are in square units. Slide 5 / 11 Slide 54 / 11 Find the total area of the rectangles. 5 Multiplying Polynomials 6 ( + 6) (5 + 8 ) = (5 + 8) + 6 (5 + 8) = (5) + (8) + 6(5) + 6(8) Return to Table of ontents 8 = 10 + 16 + 0 + 48 = 148 sq.units rea of the big rectangle rea of the horizontal rectangles rea of each box
Slide 55 / 11 Let us observe the work from the previous example, ( + 6) (5 + 8 ) Find the total area of the rectangles. 4 x = (5 + 8) + 6 (5 + 8) x = (5) + (8) + 6(5) + 6(8) = 10 + 16 + 0 + 48 = 148 Slide 56 / 11 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 57 / 11 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 : (x + 4y)(x + y) Slide 58 / 11 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: x(x + y) + 4y(x + y) x(x) + x(y) + 4y(x) + 4y(y) 6x + 4xy + 1xy + 8y 6x + 16xy + 8y First Outer Inner Inner Terms Last Terms Last Example : Slide 59 / 11 Slide 60 / 11 Try it!find each product. Try it!find each product. 1) (x - 4)(x - ) - x - 8) ) (x + )(x ) (x - y)(4x + 5y) - x y - 15y 8x Slide to check. - x + 4) 4) (x + x - 6)(x x4 + x - 8x + 4x - 4 x Slide - 7x +to1 check. Slide to check. x + x - 14x - 16 Slide to check.
Slide 61 / 11 Slide 6 / 11 8What is the total area of the rectangles shown? 4x 5 9 x 4 Slide 6 / 11 Slide 64 / 11 0 1 Slide 65 / 11 Slide 66 / 11 Find the area of a square with a side of
Slide 67 / 11 Slide 68 / 11 4What is the area of the rectangle (in square units)? x + 5x + How would we find the area of the shaded region? x + 6x + x - 6x + x - 5x + Shaded rea = Total area - Unshaded rea sq. units Slide 69 / 11 Slide 70 / 11 5What is the area of the shaded region (in sq. units)? 6What is the area of the shaded region (in square units)? 11x + x - 8 x - x - 8 7x + x - 9 x - 4x - 6 7x - x - 10 x - 10x - 8 11x - x - 8 x - 6x - 4 Slide 71 / 11 Slide 7 / 11 Square of a Sum Special inomial Products (a + b) (a + b)(a + b) a + ab + b The square of a + b is the square of a plus twice the product of a and b plus the square of b. Return to Table of ontents Ex am ple: (5x + ) (5x + )(5x + ) 5x + 0x + 9
Slide 7 / 11 Slide 74 / 11 Square of a ifference Product of a Sum and a ifference (a + b)(a - b) a + - ab + ab + - bnotice the - aband ab (a - b) (a - b)(a - b) a - ab + b a - b equals 0. The product of a + b and a - b is the square of a m inus the square of b. The square of a - b is the square of a m inus twice the product of a and b plus the square of b. Ex am ple: (y - 8)(y + 8) Rem em ber the inner and 9y - 64outer term s equals 0. (7x - 4) (7x - 4)(7x - 4) 49x - 56x + 16 Ex am ple: Slide 75 / 11 Slide 76 / 11 7 Try It! Find each product. x + 5 1. (p + 9) 9pSlide + 54p + 81 to check. x + 10x + 5 x - 10x + 5. (6 - p). (x - )(x + ) x - 5 6Slide - 1p +p to check. Slide 4x to- check. 9 Slide 77 / 11 Slide 78 / 11 8 9What is the area of a square with sides x + 4?
Slide 79 / 11 Slide 80 / 11 40 Solving Equations Return to Table of ontents Slide 81 / 11 Slide 8 / 11 Given the following equation, what conclusion(s) can be drawn? ab = 0 Zero Product Property Rule: If ab=0, then either a=0 or b=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. Slide 8 / 11 Slide 84 / 11 Given the following equation, what conclusion(s) can be drawn? (x - 4)(x + ) = 0 Since the product is 0, one of the factors m ust be 0. Therefore, either x - 4 = or 0 x + =.0 x - 4 = 0 or x + = 0 - - + 4 + 4 x = 4 or x = - Therefore, our solution set is {-, 4}. To verify the results, substitute each (x -the 4)(xoriginal + ) = equation. 0 To check x = 4: (x - 4)(x + ) = 0 solution into To check xback = - : (- - 4)(- + ) = 0 (- 7)(0) = 0 0= 0 (4-4)(4 + ) = 0 (0)(7) = 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 85 / 11 Slide 86 / 11 41Solve (a + )(a - 6) = 0. {, 6} {-, -6} 4Solve (a - )(a - 4) = 0. {, 4} {-, -4} {-, 6} {-, 4} {, -6} {, -4} Slide 87 / 11 Slide 88 / 11 4Solve (a - 8)(a + 1) = 0. {-1, -16} {-1, 16} {-1, 4} {-1, -4} Factors and Greatest ommon Factors Return to Table of ontents Slide 89 / 11 Factors of 10 Factors Unique to 10 Factors of 15 Factors Unique to 15 Factors 10 and 15 have in common Slide 90 / 11 Number ank Factors of 1 Factors of 18 Number ank 1 11 1 11 1 1 1 1 4 5 6 7 8 9 10 14 15 4 5 6 7 8 9 10 14 15 What is the greatest common factor (GF) of 10 and 15? 16 17 18 19 0 Factors Unique to 1 Factors Unique to 18 Factors 1 and 18 have in common 16 17 18 19 0 What is the greatest common factor (GF) of 1 and 18?
Slide 91 / 11 Slide 9 / 11 44 What is the GF of 1 and 15? 45 What is the GF of 4 and 48? Slide 9 / 11 Slide 94 / 11 46 What is the GF of 7 and 54? 47 What is the GF of 70 and 99? Slide 95 / 11 Slide 96 / 11 Variables also have a GF. 48 What is the GF of 8, 56 and 4? 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
Slide 97 / 11 49 What is the GF of and Slide 98 / 11? 50 What is the GF of? and Slide 99 / 11 51 What is the GF of and and Slide 100 / 11? 5 What is the GF of and and? Slide 101 / 11 Slide 10 / 11 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. Factoring out GFs Example 1 Factor each polynomial. a) 6 x4-1 5 x + x Find the GF GF: x x Return to Table of ontents 6x 4 15x x x x x Reduce each term of the polynomial dividing by the GF x ( x - 5x + 1)
Slide 10 / 11 Slide 104 / 11 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. Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has common a binomial factor. Example Factor each polynom ial. a) b) 4 m n - 7 m n Find the GF y(y - ) + 7(y - ) Find the GF (y - ) GF: y - GF: mn ( y(y - ) (y - ) + 7(y - ) (y - ) ( Example 1 Factor each polynomial. Reduce each term of the polynomial dividing by the GF Reduce each term of the polynomial dividing by the GF (y - )(y + 7) m n(4 n - 7 n) Slide 105 / 11 Slide 106 / 11 Sometimes the distributive property can be used to factor a polynomial that is not in simplest form but has common a binomial factor. In working with common binomial factors, look for factors that are opposites of each other. For example: (x - y) = - (y - x ) because Example Factor each polynom ial. x - y = x + (- y) = - y + x = - (y - x) b) Find the GF GF: Reduce each term of the polynomial dividing by the GF Slide 107 / 11 Slide 108 / 11 5 True or False: y - 7 = -1( 7 + y) True False 54 True or False: 8 - d = -1( d + 8) True False
Slide 109 / 11 Slide 110 / 11 55 True or False: 8c - h = -1( -8c + h) True 56 True or False: -a - b = -1( a + b) True False False Slide 111 / 11 Slide 11 / 11 In working with common binomial factors, look for factors that are opposites of each other. For example: (x - y) = - (y - x ) because In working with common binomial factors, look for factors that are opposites of each other. For example: x - y = x + (- y) = - y + x = - (y - x) (x - y) = - (y - because x) x - y = x + (- y) = - y + x = - (y - x) Example Factor each polynomial. Example Factor each polynomial. a) b) n(n - ) - 7( - n) p(h - 1) + 4(1 - h) Find the GF Find the GF GF: Reduce each term of the polynomial dividing by the GF GF: Reduce each term of the polynomial dividing by the GF (n - )(n + 7) (h - 1)(p - 4) Slide 11 / 11 Slide 114 / 11 57 If possible, Factor 58 If possible, Factor lready Simplified lready Simplified
Slide 115 / 11 Slide 116 / 11 59 If possible, Factor 60 If possible, Factor lready Simplified lready Simplified Slide 117 / 11 Slide 118 / 11 61 If possible, Factor lready Simplified Identifying & Factoring: x + bx + c Return to Table of ontents Slide 119 / 11 Slide 10 / 11 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 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. Ex am ples: hoose all of the description that apply. ubic Quadratic Linear onstant Trinomial inomial Monomial
Slide 11 / 11 Slide 1 / 11 6 hoose all of the descriptions that apply to: Quadratic 6 hoose all of the descriptions that apply to: Quadratic Linear Linear onstant onstant Trinomial Trinomial E inomial E inomial F Monomial F Monomial Slide 1 / 11 Slide 14 / 11 64 hoose all of the descriptions that apply to: Quadratic 65 hoose all of the descriptions that apply to: Quadratic Linear Linear onstant onstant Trinomial Trinomial E inomial E inomial F Monomial F Monomial Slide 15 / 11 Slide 16 / 11 Simplify. nswer ank 1 ) (x + )(x + ) = ) (x - 4 )(x - 1 ) = To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: x - 5 x + 4 Factors of 6 have the same signs. x - 4 x - 5 ) (x + 1 )(x - 5 ) = x + 5 x + 6 4 ) (x + 6 )(x - ) = x + 4 x - 1 Slide each polynomial from the circle to the correct expression. RELL What did we do?? Look for a pattern!! Factors of 6 Sum to 5? 1, 6 7, 5 Factors of 6 add to +5. oth factors must be positive.
Slide 17 / 11 Slide 18 / 11 Examples: To Factor a Trinomial with a Lead oefficient of 1 (x - 8)(x - 1) Recognize the pattern: Factors of 6 have the same signs. Factors of 6 Sum to -7? -1, -6-7 -, - -5 Factors of 6 add to -7. oth factors must be negative. Slide 19 / 11 Slide 10 / 11 66 The factors of 1 will have what kind of signs given the following equation? oth positive 67 The factors of 1 will have what kind of signs given the following equation? oth positive oth Negative oth negative igger factor positive, the other negative igger factor positive, the other negative The bigger factor negative, the other positive The bigger factor negative, the other positive Slide 11 / 11 Slide 1 / 11 68 Factor (x + 1)(x + 1) 69 Factor (x + 1)(x + 1) (x + 6)(x + ) (x + 6)(x + ) (x + 4)(x + ) (x + 4)(x + ) (x - 1)(x - 1) (x - 1)(x - 1) E (x - 6)(x - 1) E (x - 6)(x - 1) F (x - 4)(x - ) F (x - 4)(x - )
Slide 1 / 11 Slide 14 / 11 70 Factor 71 Factor (x + 1)(x + 1) (x + 1)(x + 1) (x + 6)(x + ) (x + 6)(x + ) (x + 4)(x + ) (x + 4)(x + ) (x - 1)(x - 1) (x - 1)(x - 1) E (x - 6)(x - 1) E (x - 6)(x - ) F (x - 4)(x - ) F (x - 4)(x - ) Slide 15 / 11 Slide 16 / 11 To Factor a Trinomial with a Lead oefficient of 1 To Factor a Trinomial with a Lead oefficient of 1 Recognize the pattern: Recognize the pattern: Factors of 6 have the opposite signs. Factors of 6 have the opposite signs. Factors of 6 Sum to -5? 1, -6-5, - -1 Factors of 6 add to -5. Larger factor must be negative. Factors of 6 Sum to 1? -1, 6 5 -, 1 Slide 17 / 11 Examples Factors of 6 add to +1. Larger factor must be positive. Slide 18 / 11 7 The factors of -1 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 19 / 11 Slide 140 / 11 7 The factors of -1 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 74 Factor (x + 1)(x - 1) (x + 6)(x - ) (x + 4)(x - ) (x - 1)(x + 1) E (x - 6)(x + 1) F (x + 4)(x - ) Slide 141 / 11 Slide 14 / 11 75 Factor 76 Factor (x + 1)(x - 1) (x + 1)(x - 1) (x + 6)(x - ) (x + 6)(x - ) (x + 4)(x - ) (x + 4)(x - ) (x - 1)(x + 1) (x - 1)(x + 1) E (x - 6)(x + 1) E (x - 6)(x + 1) F unable to factor using this method F (x - 4)(x + ) Slide 14 / 11 Slide 144 / 11 77 Factor the following Mixed Practice (x - )(x - 4) (x + )(x + 4) (x - )(x +4) (x + )(x - 4)
Slide 145 / 11 Slide 146 / 11 78 Factor the following 79 Factor the following (x - )(x - 5) (x - )(x - 4) (x + )(x + 4) (x + )(x + 5) (x - )(x +5) (x +)(x +6) (x + )(x - 5) (x + 1)(x+1) Slide 147 / 11 Slide 148 / 11 Steps for Factoring a Trinomial 80 Factor the following (x - )(x - 5) (x + )(x + 5) (x - )(x +5) (x + )(x - 5) 1) See if a monomial can be factored out. ) Need numbers that multiply to the constant ) and add to the middle number. 4) Write out the factors. There is no common STEP monomial,so factor: 1 Slide 149 / 11 Steps for Factoring a Trinomial STEP STEP STEP STEP 4 Slide 150 / 11 Factor: 1) See if a monomial can be factored out. ) Need numbers that multiply to the constant ) and add to the middle number. 4) Write out the factors. There is no common STEP monomial,so factor:1 STEP Factor out STEP 1 STEP 4 STEP STEP Steps for Factoring a Trinomial 1) See if a monomial can be factored out. ) Need numbers that multiply to the constant ) and add to the middle number. 4) Write out the factors. STEP 4
Slide 151 / 11 Slide 15 / 11 81 Factor completely: Factor: Factor out STEP 1 STEP STEP STEP 4 Steps for Factoring a Trinomial 1) See if a monomial can be factored out. ) Need numbers that multiply to the constant ) and add to the middle number. 4) Write out the factors. Slide 15 / 11 Slide 154 / 11 8 Factor completely: 8 Factor completely: Slide 155 / 11 Slide 156 / 11 84 Factor completely: 85 Factor completely:
Slide 157 / 11 Slide 158 / 11 Factoring Using Special Patterns 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. Return to Table of ontents Slide 159 / 11 Perfect Square Trinomials The Square of a Sum and the Square of a difference have products that are called Perfect Square Trinomials. Slide 160 / 11 Examples of 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 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 161 / 11 Is the trinomial a perfect square? rag the Perfect Square Trinomials into the ox. Slide 16 / 11 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) Examples: Only Perfect Square Trinomials will remain visible.
Slide 16 / 11 Slide 164 / 11 86 Factor 87 Factor Not a perfect Square Trinomial Not a perfect Square Trinomial Slide 165 / 11 Slide 166 / 11 ifference of Squares 88 Factor The Product of a Sum and a ifference is a difference of Squares. Not a perfect Square Trinomial ifference of Squares is recognizable by seeing each term in the binomial are perfect squares and the operation is subtraction. Slide 167 / 11 Slide 168 / 11 Is the binomial a difference of squares? Examples of ifference of Squares rag the ifference of Squares binomials into the ox. Only ifference of Squares will remain visible.
Slide 169 / 11 Slide 170 / 11 Factoring a ifference of Squares 89 Factor Once a binomial is determined to be a ifference of Squares, it factors following the pattern: (sq rt of 1st term - sq rt of nd term)(sq rt of 1st term + sq rt of nd term) Examples: Not a ifference of Squares Slide 171 / 11 Slide 17 / 11 90 Factor 91 Factor Not a ifference of Squares Not a ifference of Squares Slide 17 / 11 Slide 174 / 11 9 Factor 9 Factor using ifference of Squares: Not a ifference of Squares
Slide 175 / 11 Slide 176 / 11 How to factor a trinomial of the form ax² + bx + c. Example: Factor d² + 15d + 18 18 = 6 Factoring Trinomials: ax + bx + c Now find two integers whose product is 6 and whose sum is equal to b or 15. Factors of 6 Sum = 1 5? 1 + 6 = 7 1, 6 + 18 = 0, 18 + 1 = 15, 1 Now substitute 1 + into the equation for 1 5. d² + (1 + )d + 1 8 istribute d² + 1 d + d + 1 8 Group and factor GF d(d + 6 ) + (d + 6 ) Factor common binomial (d + 6 )( d + ) Remember to check using FOIL! Return to Table of ontents Slide 177 / 11 Slide 178 / 11 Factor. 1 5 x² - 1 x + Factor. b - b- 1 0 a = 1 5 and c =, but b = - 1 Since both a and c are positive, and b is negative we need to find two negative factors of 0 that add up to - 1 a =, 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 - 0 and that add up to - 1. Since the sum is negative, larger factor of - 0 must be negative. Factors of - 0 Sum = - 1? Sum = - 1? Factors of 0-1,,, 5, - 0 15 10 6-1 5 + + + + - 0 = 15 = 10 = 6 = - - 1-17 - 1 11 1, - 0, - 10 4, - 5 Slide 179 / 11 1 + - 0 = - 19 + - 10 = - 8 4 + -5 = -1 Slide 180 / 11 Factor 6 y² - 1 y - 5 polynomial that cannot be written as a product of two polynomials is called aprime polynomial.
Slide 181 / 11 Slide 18 / 11 94 Factor 95 Factor Prime Polynomial Prime Polynomial Slide 18 / 11 Slide 184 / 11 96 Factor Factoring 4 Term Polynomials Prime Polynomial Return to Table of ontents Slide 185 / 11 Polynom ials with four term s likeab - 4b + 6a - 4, can be factored by grouping term s of the polynom ials. Example 1 : ab - 4 b + 6 a - 4 (ab - 4b) + (6a - 4) 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! Slide 186 / 11 Example : 6x y + 8x - 1y - 8 (6x y + 8x ) + (- 1y - Group 8) x (y + 4) + (- 7)(y +Factor 4) GF (y + 4) (x - 7)Factor com m on binom ial
Slide 187 / 11 Slide 188 / 11 You m ust be able to recognize additive inverses!!! ( - aand a - are additive inverses because their sum is equal to zero.) Rem em ber - a = - 1(a -. ) Example : 15x - x y + 4y - 0 (15x - x y) + (4y - 0) Group x (5 - y) + 4(y - Factor 5) GF x (- 1)(- 5 + y) + 4(y -Notice 5) additive inverses - x (y - 5) + 4(y -Sim 5) plify (y - 5) (- x + 4)Factor com m on binom ial 97 Factor 15ab - a + 10b - (5b - 1)(a + ) (5b + 1)(a + ) (5b - 1)(a - ) (5b + 1)(a - 1) Remember to check each problem by using FOIL. Slide 189 / 11 Slide 190 / 11 98 Factor 10mn - 5mn + 6m - 15 99 Factor 0ab - 5b - 6 +6a (4a - 7)(5b - 9) (m-5)(5mn+) (4a - 7)(5b + 9) (m+5)(5mn-) (4a + 7)(5b - 9) (m+5)(5mn+) (4a + 7)(5b + 9) (m-5)(5mn-) Slide 191 / 11 Slide 19 / 11 100 Factor a - ab + 7b - 7a (a - b)(a - 7) (a - b)(a + 7) (a + b)(a - 7) (a + b)(a + 7) Mixed Factoring Return to Table of ontents
Slide 19 / 11 Slide 194 / 11 Summary of Factoring Examples Factor the Polynomial r - 9r + 6r r(r - r + ) r(r - 1)(r - ) Factor out GF 4 Terms Terms Terms ifference of Squares 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 195 / 11 Slide 196 / 11 101 Factor completely: 10 Factor completely prime polynomial Slide 197 / 11 Slide 198 / 11 10 Factor 104 Factor completely prime polynomial 10w (x -10x +100)
Slide 199 / 11 Slide 00 / 11 105 Factor Solving Equations by Factoring Prime Polynomial Return to Table of ontents Slide 01 / 11 Slide 0 / 11 Recall ~ Given the following equation, what conclusion(s) can be Given the following equation, what conclusion(s) can be drawn? (x - 4)(x + ) = 0 drawn? Since the product is 0, one of the factors m ust be 0. Therefore, either x - 4= 0 or x + =.0 ab = 0 x - 4 = 0 or + 4 + 4 x = 4 or Since the product is 0, one of the factors, a or b, m ust be 0. This is known as the Zero Product Property. x + = 0 - - x = - Therefore, our solution set is {-, 4}. To verify the results, substitute each solution back into the original equation. To check x = - :(x - 4)(x + ) = 0 To check x = 4: (x - 4)(x + ) = 0 (- - 4)(- + ) = 0 (- 7)(0) = 0 0= 0 Slide 0 / 11 What if you were given the following equation? Slide 04 / 11 Solve 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} Recall the Steps for Factoring a Trinomial 1) See if a monomial can be factored out. ) Need numbers that multiply to the constant ) and add to the middle number. 4) Write out the factors. Now... 1) Set each binomial equal to zero. ) Solve each binomial for the variable. (4-4)(4 + ) = 0 (0)(7) = 0 0= 0
Slide 05 / 11 Slide 06 / 11 106 hoose all of the solutions to: 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 E F Slide 07 / 11 Slide 08 / 11 107 hoose all of the solutions to: 108 hoose all of the solutions to: E E F F Slide 09 / 11 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 + 0t. When will the rocket hit the ground? 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 0 seconds. The 0 is an extraneous (extra) answer. Slide 10 / 11 109 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
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