Chapter 8: Factoring

Chapter 8 : Factoring Fill in the titles on the foldable Greatest Common Factor (GCF) 8.1 Grouping 8.2 Trinomials – x2 + bx + c 8.3 Trinomials – ax2 + bx + c 8.4 Differences of Squares 8.5 Perfect Squares 8.6 Sums and Differences of Cubes Quadratic Form Combinations

8.2 Greatest Common Factor (top) Find the GCF of the terms Write the GCF then the remaining part of each term in parentheses

8.2 Greatest Common Factor (bottom) Ex: 12a2 + 16a Ex: 3p2q – 9pq2 + 36pq 2 x 2 x 3 x a x a x 3 x p x p x q = 2 x 2 x a =4a -1 x 3 x 3 x p x q x q 2 x 2 x 2 x 2 x a 3 x p x q = 3pq 2 x 2 x 3 x 3 x p x q 4a(3a + 4) 3pq(p - 3q + 12)

Your Turn – What is the Greatest Common Factor (GCF) Ex: 15x + 20 Ex: 8x³y + 2x²y² - 4 xy³ GCF=5 2 x x x y = 2xy =5(3x + 4) =2xy(4x² + xy – 2y²)

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

8.2 Factor by Grouping (bottom) Ex: 4ab + 8b + 3a + 6 Ex: 3p – 2p2 – 18p + 27 (4ab + 8b)( +3a + 6) (3p – 2p2 )( – 18p + 27) 2 x 2 x a x b 3 x a = 3 =4b 3 x p 2 x 2 x 2 x b 2 x 3 = p -1 x 2 x 3 x 3 x p -1 2 x p x p = 9 3 x 3 x 3 4b(a + 2) +3 (a + 2) p(3 – 2p) + 9(-2p + 3) (4b + 3)(a + 2) (p + 9)(-2p + 3)

Your turn- Factor by Grouping Ex: 5x(x – 2) + 6(x – 2) Ex: 3x² - 2x + 6x – 4 (3x² – 2x )+( 6x - 4) x(3x – 2) + 2(3x - 2) (x – 2) (5x + 6) (x + 2)(3x – 2)

Factoring Trinomials – x2 + bx + c Get everything on one side (equal to zero) Split into two groups ( )( ) = 0 Factor the first part x2 (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) Use Distributive Property (Foil) to check your answer Use Zero Product Property to solve if needed

Factoring Trinomials – x2 + bx + c Ex: x2 + 6x + 8 Ex: r2 – 2r - 24 Ex: s2 – 11s + 28 = 0 8 1, 8 2, 4 (x )(x ) 24 1, 24 2, 12 3, 8 4, 6 28 1, 28 2, 14 4, 7 (r )(r ) (s )(s ) (x + 2)(x + 4) (s- 4)(s - 7) = 0 (r + 4)(r - 6) Check your work Check your work Check your work FOIL s2 – 7s – 4s + 28 s2 – 11s + 28 FOIL x2 + 2x + 4x + 8 x2 + 6x + 8 FOIL r2 – 6r + 4r - 24 r2 - 2x - 24 s – 4 = 0 s – 7 = 0 +4 +4 +7 +7 s = 4 s = 7 s = 4 and 7

8.4 Factoring Trinomials – ax2 + bx + c (top) Get everything on one side (equal to zero) Find product of the first and last parts Find the factors of the product Rewrite the ax2 then fill in the pair of factors that adds or subtracts to make the second part followed by c Factor by grouping if you can’t factor = prime (use the zero product property to solve if needed)

8.4 Factoring Trinomials – ax2 + bx + c (bottom) Hint: find the gcf to pull it out and make the numbers smaller if possible Ex: 5x2 + 13x + 6 Ex: 10y2 - 35y + 30 = 0 2 x 6 = 12 5 x 6 = 30 5(2y2 - 7y + 6) = 0 1, 12 2, 6 3, 4 ( ) ( ) 5x2 + 3x + 10x + 6 1, 30 2, 15 3, 10 5, 6 5[(2y2 -3y)(-4y + 6)]=0 x(5x + 3) + 2(5x + 3) 5[y(2y - 3)-2(2y - 3)]=0 (x + 2)(5x + 3) 5(y - 2)(2y - 3) = 0 Solve for y. y – 2 = 0 2y – 3 = 0 y = 2 and 1.5

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: n2 - 25 n x n 5 x 5 (n + 5)(n - 5)

8.5 Factoring Differences of Squares (bottom) Ex: 5x3 + 15x2 – 5x - 15 Ex: 121a = 49a3 -121a -121a 5[x3 + 3x2 – x – 3] 0 = 49a3 – 121a 5[ (x3 + 3x2)( – x – 3)] 0 = a(49a2 – 121) 5[ x2(x + 3) - 1(x + 3)] 0 = a(7a x 7a 11 x 11) 0 = a(7a + 11)(7a - 11) 5[(x2 – 1)(x + 3)] a = 0 7a + 11 = 0 7a - 11 = 0 5[(x x x 1 x 1)(x + 3)] -11 -11 +11 +11 7a = -11 7a = 11 5(x + 1)(x - 1)(x + 3) /7 /7 /7 /7 a= -11/7 a = 11/7 a = -11/7, 0, and 11/7

8.6 Factoring Perfect Squares (top) Perfect Square Trinomial: Is the first term a perfect square? Is the last term a perfect square? Does the second term = 2 x the product of the roots of the first and last terms? The third term (c) must be positive Use the sign of the second term If any of these answers is no- it is not a perfect square trinomial

8.6 Factoring Perfect Squares (bottom) Ex: x2 – 14x + 49 Ex: a2 – 8a - 16 x x x 7 x 7 a x a 4 x 4 2 x x x 7= 14x 4 x 4 = 16 but it is a negative 16 so it can’t be a perfect square (x – 7)2 Ex: 9y2 + 12y + 4 1. 9y2 = 3y x 3y yes 2. 4 = 2 x 2 yes 3. 2(3y x 2) = 2(6y) = 12y yes (3y + 2)2

Sum and Difference of Cubes (top) a3 + b3 = (a + b)(a2 – ab + b2) a3 - b3 = (a - b)(a2 + ab + b2) Same - Opposite - Always Positive To remember the signs: SOAP

Sum and Difference of Cubes (bottom) Ex: x3 + 8 Ex: 27x3 – 64y3 Ex: 1000y3 – 216 Ex: 125a3 + 27b3 x x x 2 2 2 3x 3x 3x 4y 4y 4y x2 - 1x2x + 22 (3x - 4y) (9x2 + 12xy + 16y2) (x + 2) (x2 – 2x + 4) (10y - 6) (100x2 + 60y + 36) (5a + 3b) (25a2 – 15ab + 9b2)

Quadratic Form (Bottom) Ex: x4 + 3x2 + 2 Ex: x4 – 16 Ex: x - + 21 Ex: x4 + 7x2 + 12 (x2 )(x2 ) (x2 )(x2 ) (x2 + 4)(x2 - 4) (x2 + 1)(x2 + 2) (x2 + 1)(x2 + 2)(x2 – 2) ( - 3)( - 7) (x2 )(x2 ) (x2 + 3)(x2 + 4)

Combinations of Factoring Types (Top) First look for the GCF and factor out if possible Next look for patterns (perfect squares, difference of squares or sum and difference of cubes) If no patterns appear factor the trinomial like normal

Combinations of Factoring Types (bottom) Ex: 4x2 – 100 Ex: 3x2 – 3x – 60 Ex: 8x6 – 64x3 Ex: 8x3 – 32x 4(x2 – 25) 3(x2 – x - 20) 20 1, 20 2, 10 4, 5 4(x + 5)(x – 5) (x + 4)(x – 5) 8x3 (x3 – 8) 8x(x2 – 4) 8x3(x – 2)(x2 + 2x + 4) 8x(x + 2)(x – 2)