5.5: Factoring the Sum and Difference of Two Cubes

Objectives Factor the sum of two cubes. Factor the difference of two cubes. Factor a polynomial involving the sum or difference of two cubes.

Factoring the sum and difference of two cubes Recall: x2 – y2 = (x + y)(x – y) In this section, we will look at x3 + y3 x3 – y3

Factor the sum and difference of two cubes We note: (x + y)(x2 – xy + y2) = x3 – x2y + xy2 + x2y– xy2 + y3 = x3 + y3 (x - y)(x2 + xy + y2) = x3 + x2y + xy2 - x2y -xy2 -y3 = x3 - y3 So, we can conclude: x3 + y3 = (x + y)(x2 – xy + y2) x3 - y3 = (x - y)(x2 + xy + y2)

Example Useful to know cubes of 1, 2, 3, 4, etc. Factor: x3 + 8 Solution: x3 + 8 = x3 + 23 = (x + 2)(x2 – x  2 + 22) = (x + 2)(x2 – 2x + 4) n 1 2 3 4 5 6 7 8 9 10 n3 27 64 125 216 343 512 729 1000

Example Factor: a3 – 64b3 Solution: a3 – 64b3 = a3 – (4b)3 = (a – 4b)(a2 + a(4b) + (4b)2) = (a – 4b)(a2 + 4ab + 16b2)

Example Factor: –2t 5 + 128t 2 Solution: –2t 5 + 128t 2 = –2t 2(t 3 – 64) = –2t 2(t 3 – 43) = –2t 2(t – 4)(t 2 + 4t + 16)

Your Turn Factor: –2x 5 + 54x 2 (Hint: First, factor the GCF) Solution: –2x5 – 54x 2 = –2x2(x 3 + 27) = –2x 2(x 3 + 33) = –2x 2(x + 3)(x 2 – (3)x + 32) = –2x 2(x + 3)(x 2 – 3x + 9)