Factoring Polynomials Objective: To factor polynomials by GCF, special products and grouping terms

By GCF Factor each Polynomial 3x4 – 12x3 + 6x2 The GCF of 3x4, -12x3 and 6x2 is 3x2 3x4 – 12x3 + 6x2 = 3x2(x2 – 4x + 2) 8xy2 – 10x2y The GCF of 8xy2 and -10x2y is 2xy 8xy2 – 10x2y = 2xy(4y – 5x)

Perfect Square Trinomials a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 Example: factor 4x2 – 20x + 25 (2x)2 – 2(2x)(5) + (5)2 (2x – 5)2 Factor x2 + 12x + 36 (x)2 + 2(x)(6) + (6)2 (x + 6)2

Difference Of Two Squares a2 – b2 = (a + b)(a – b) Example: factor x2 – 49 (x)2 – (7)2 = (x + 7)(x – 7) Factor: 25h2 – 9k2 (5h)2 – (3k)2 = (5h + 3k)(5h – 3k)

Sum And Difference Of Cubes a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) Example: factor x3 – 8 (x)3 – (2)3 = (x – 2)(x2 + (x)(2) + (2)2) = (x – 2)(x2 + 2x + 4) Factor x3 + 27y3 (x)3 + (3y)3 = (x +3y)(x2 – x(3y) + (3y)2) = (x + 3y)(x2 – 3xy +9y2)

Factoring By Grouping In many instances a polynomial will not be a special product, but may be able to be factored by rearranging its terms. Factor 2xy – 3 – x + 6y 2xy – x + 6y – 3 x(2y – 1) + 3(2y – 1) (2y – 1)(x + 3) Note: this process often requires some trial and error.

Try These! Factor each Polynomial 12x3y2 – 6x2y +9xy3 x2 – 8x + 16 x2 – 64y2 8x3 + 125y3 8x2y + 12x + 3y + 2xy2 3xy(4x2y – 2x + 3y2) (x – 4)2 (x – 8y)(x + 8y) (2x + 5y)(4x2 – 10xy + 25y2) (4x + y)(2xy + 3)

Try These! All these problems will require more than one type of factorization x4 – 16 x6 – y6 2x2y – 8y3 6x2y + 3xy2 – 6xy – 3y2 (x2 + 4)(x + 2)(x – 2) solution (x + y)(x – y)(x2 + xy + y2)(x2 – xy + y2) solution 2y(x + 2y)(x – 2y) solution 3y(2x + y)(x – 1) solution

x4 – 16 (x2)2 – 42 ((x2) + 4)((x2) – 4) (x2 + 4)((x)2 – (2)2)

x6 – y6 (x3)2 – (y3)2 (x3 + y3)(x3 – y3) (x + y)(x2 – xy + y2)(x – y)(x2 + xy +y2)

2x2y – 8y3 2x2y – 23y3 2y(x2 – 22y2) 2y((x)2 – (2y)2) 2y(x + 2y)(x – 2y)

6x2y + 3xy2 – 6xy – 3y2 3xy(2x + y) – 3y(2x + y) (2x + y)(3xy – 3y) 3y(x – 1)(2x + y)