Factoring Sums and Differences Section 9.8

Sums and Differences of Cubes x3 + y3 = (x + y) (x2 – xy + y2) x3 - y3 = (x - y) (x2 + xy + y2)

For general odd powers… xn + yn = (x + y) (xn-1 – xn-2 y + xn-3y2 - … - xyn-2 + yn-1) xn - yn = (x - y) (xn-1 + xn-2 y + xn-3y2 + … + xyn-2 + yn-1)

Find the fourth roots of 16. Example 1 Find the fourth roots of 16. x4 - 16 = 0 (x2 + 4)(x2 – 4) x = 2i , x = -2i x = 2 , x = -2

Example 2 Factor x3 – 64y12 over the set of polynomial with rational coefficients (x – 4y4)(x2 + 4xy4 + 16y8)

Factor m6 + n6 over the set of polynomial with rational coefficients. Example 3 Factor m6 + n6 over the set of polynomial with rational coefficients. Same as (m2 + n2)3 (m2 + n2)(m4 – m2n2 + n4)

Example 4 Factor x10 – y10 completely over the set of polynomials with rational coefficients. Same as (x5 – y5)2 (x5 – y5)(x5 + y5) (x – y)(x4 +x3y + x2y2 + xy3 + y4) (x + y )(x4 - x3y + x2y2 - xy3 + y4)

Homework Pages 605 – 606 4 - 7, 11 - 19