# chapter 8: factoring. prime factoring & factor a monomial 8.1 greatest common factor (gcf) 8.1...

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Chapter 8: Factoring Slide 2 Prime Factoring & Factor a monomial 8.1 Greatest Common Factor (GCF) 8.1 Factor Using Distributive Property 8.2 Factory by Grouping 8.2 Zero Product Property 8.2 Factoring Trinomials x 2 + bx + c 8.3 Factoring Trinomials ax 2 + bx + c 8.4 Factoring Differences of Squares 8.5 Factoring Perfect Squares 8.6 Square Root Property 8.6 Rational Expressions 11.2 Fill in the titles on the foldable Slide 3 8.1 Prime factoring and factor a monomial (top) Prime # = factors only include 1 and itself Composite # = more than two factors Ex: Prime factor 90Prime numbers: 1, 3, 5, 7, 11, 13, 17, 23, 29, 31, 37 . 90 245 315 35 2 x 3 x 3 x 5 = 2 x 3 2 x 5 Slide 4 8.1 Prime factoring and factor a monomial (bottom) Factor a monomial = list all factors separately (no exponents) Ex: 38rs 2 t Ex: -66pq 2 219 38 2 x 19 x r x s x s x t -66 66 233 311 -1 x 2 x 3 x 11 x p x q x q Slide 5 8.1 Greatest Common Factor (top) GCF = the largest factor that is in all the given monomials 1. factor all monomials 2. circle all common factors 3. Multiply all the circled factors Slide 6 8.1 Greatest Common Factor (bottom) Ex: 84 & 70Ex: 36x 2 y & 54xy 2 z 84 242 221 37 2 x 2 x 3 x 7 70 235 57 2 x 5 x 7 2 x 7= 14 36 218 29 33 2 x 2 x 3 x 3 x x x x x y 54 227 39 33 2 x 3 x 3 x 3 x x x y x y x z 2 x 3 x 3 x x x y= 18xy Slide 7 8.2 Factor Using Distributive Property (top) Find the GCF of the monomials Write each term as a product of the GCF and the remaining factors Simplify the remaining factors Slide 8 8.2 Factor Using Distributive Property (bottom) Ex: 12a 2 + 16aEx: 3p 2 q 9pq 2 + 36pq 12 26 2 3 16 28 24 22 2 x 2 x 3 x a x a 2 x 2 x 2 x 2 x a = 2 x 2 x a=4a 4a(3a) +4a(4) = 4a(3a + 4) -9 9 x 3 x p x p x q 36 218 29 33 -1 x 3 x 3 x p x q x q 2 x 2 x 3 x 3 x p x q 3 x p x q =3pq 3pq(p) +3pq(-3q) +3pq(12) = 3pq(p - 3q + 12) 33 Slide 9 8.2 Factor by Grouping (top) Group the terms (first two and last two) Find the GCF of each group Write each group as a product of the GCF and the remaining factors Combine the GCFs in a group and write the other group as the second factor Slide 10 8.2 Factor by Grouping (bottom) Ex: 4ab + 8b + 3a + 6 (4ab + 8b)(+ 3a + 6) 4 22 8 24 22 2 x 2 x a x b 2 x 2 x 2 x b =4b 4b(a + 2) 3 x a 6 23 2 x 3 = 3 +3 (a + 2) (4b + 3)(a + 2) Ex: 3p 2p 2 18p + 27 (3p 2p 2 )( 18p + 27) 3 x p -1 2 x p x p = p -18 29 33 -1 x 2 x 3 x 3 x p 18 27 39 33 3 x 3 x 3 = 9 p(3 2p)+ 9(-2p + 3) (p + 9)(-2p + 3) Slide 11 8.2 Zero Product Property (top) Roots = the solutions to the equation When an equation is factored and equal to zero: Set each factor equal to zero and solve for the variable Slide 12 8.2 Zero Product Property (bottom) Ex: (d 5)(3d + 4) = 0 d 5 = 03d + 4 = 0 + 5 + 5 d = 5 - 4 - 4 3d = -4 /3 d = -4/3 Roots are d = 5 and -4/3 Ex: 7f 2 35f = 0 7 x f x f -35 35 5 7 -1 x 5 x 7 x f = 7f 7f(f)+ 7f(-5) 7f(f 5) = 0 7f = 0 f 5 = 0 /7 f = 0 + 5 + 5 f = 5 Roots are f = 0 and 5 Slide 13 8.3 Factoring Trinomials x 2 + bx + c (top) Get everything on one side (equal to zero) Split into two groups ( )( ) = 0 Factor the first part x 2 (x )(x ) = 0 Find all the factors of the third part (part c) Fill in the factors of c that will add or subtract to make the second part (bx) Foil to check your answer Use Zero Product Property to solve if needed Slide 14 8.3 Factoring Trinomials x 2 + bx + c (bottom) Ex: x 2 + 6x + 8 (x )(x ) 8 1, 8 2, 4 (x + 2)(x + 4) FOIL x 2 + 2x + 4x + 8 x 2 + 6x + 8 Ex: r 2 2r - 24 (r )(r ) 24 1, 24 2, 12 3, 8 4, 6 (r + 4)(r - 6) FOIL r 2 6r + 4r - 24 r 2 - 2x - 24 Ex: s 2 11s + 28 = 0 (s )(s ) 28 1, 28 2, 14 4, 7 (s- 4)(s - 7) = 0 FOIL s 2 7s 4s + 28 s 2 11s + 28 s 4 = 0s 7 = 0 +4 +4 +7 +7 s = 4s = 7 s = 4 and 7 Slide 15 8.4 Factoring Trinomials ax 2 + bx + c (top) Get everything on one side (equal to zero) Put the first part in each set of parentheses Find product of the first and last parts Find the factors of the product Fill in the pair of factors that adds or subtracts to the second part Remove the GCF from one set of parentheses Write what is left of the that group as one factor and then the other group as the other factor if you cant factor = prime (use the zero product property to solve if needed) Slide 16 8.4 Factoring Trinomials ax 2 + bx + c (bottom) Ex: 5x 2 + 13x + 6Ex: 10y 2 - 35y + 30 = 0 5 x 6 = 30 1, 30 2, 15 3, 10 5, 6 (5x + 10)(5x + 3) (x + 2)(5x + 3) 5(2y 2 - 7y + 6) = 0 Hint: find the gcf to pull it out and make the numbers smaller if possible 2 x 6 = 12 1, 12 2, 6 3, 4 2 x y -1 x 2 x 2 = 2 = 5 5(y - 2)(2y - 3) = 0 y 2 = 02y 3 = 0 Solve for y. y = 2 and 1.5 (5x )(5x ) 5x: x 5 10: 2 5 (5x + 10) 5(x + 2) 5(2y )(2y )=0 5(2y - 4)(2y - 3)=0 (2y - 4) 2(y - 2) Slide 17 8.5 Factoring Differences of Squares (top) Factor each term Write one set of parentheses with the factors adding and one with the factors subtracting Foil to check your answer Ex: n 2 - 25Ex: 9x 3 4x n x n5 x 5 (n + 5)(n - 5) Hint: find the gcf to pull it out and make the numbers smaller if possible x(9x 2 4) x[ 3x x 3x2 x 2] x(3x + 2)(3x - 2) Slide 18 8.5 Factoring Differences of Squares (bottom) Ex: 5x 3 + 15x 2 5x - 15Ex: 121a = 49a 3 5[x 3 + 3x 2 x 3] 5[ (x 3 + 3x 2 )( x 3)] 3 x x x x x x x x x = x 2 5[ x 2 (x + 3) -1 x x -1 x 3 = -1 - 1(x + 3)] 5[(x 2 1)(x + 3)] 5[(x x x 1 x 1)(x + 3)] 5(x + 1)(x - 1)(x + 3) -121a -121a 0 = 49a 3 121a 0 = a(49a 2 121) 0 = a(7a x 7a 11 x 11) 0 = a(7a + 11)(7a - 11) a = 07a + 11 = 07a - 11 = 0 -11 -11 +11 +11 7a = -117a = 11 /7 a= -11/7a = 11/7 a = -11/7, 0, and 11/7 Slide 19 8.6 Factoring Perfect Squares (top) Perfect Square Trinomial: Is the first term a perfect square? Is the last term a perfect square (must be a positive number)? Does the second term = 2 x the product of the roots of the first and last terms? If any of these answers is no- it is not a perfect square trinomial Slide 20 8.6 Factoring Perfect Squares (bottom) Ex: x 2 14x + 49a 2 8a - 16 Ex: 9y 2 + 12y + 4 1. 9y 2 =3y x 3y yes 2. 4 =2 x 2 yes 3. 2(3y x 2) =2(6y) =12y yes (3y + 2) 2 (x 7) 2 7 x 7x x x 2 x x x 7= 14x 4 x 4a x a 4 x 4 = 16 but it is a negative 16 so it cant be a perfect square Slide 21 8.7 Square Root Property Ex: (y 8) 2 = 7Ex: (b 7) 2 = 36 +8 b 7 = 6b 7 = -6 +7 b = 13b = 1 Slide 22 11.2 Rational Expressions (top) You can only cancel factors if they are exactly the same groups Ex: Slide 23 11.2 Rational Expressions (bottom) Ex:

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