# How do you factor a^3b^3 - 8x^6y^9?

May 28, 2015

This is a difference of cubes:

${a}^{3} {b}^{3} - 8 {x}^{6} {y}^{9}$

$= {\left(a b\right)}^{3} - {\left(2 {x}^{2} {y}^{3}\right)}^{3}$

$= \left(\left(a b\right) - \left(2 {x}^{2} {y}^{3}\right)\right) \left({\left(a b\right)}^{2} + \left(a b\right) \left(2 {x}^{2} {y}^{3}\right) + {\left(2 {x}^{2} {y}^{3}\right)}^{2}\right)$

$= \left(a b - 2 {x}^{2} {y}^{3}\right) \left({a}^{2} {b}^{2} + 2 a b {x}^{2} {y}^{3} + 4 {x}^{4} {y}^{6}\right)$

using the identity ${A}^{3} - {B}^{3} = \left(A - B\right) \left({A}^{2} + A B + {B}^{2}\right)$