# How do you factor (A-B)^3 - B^3?

May 31, 2015

This is a difference of cubes, so we can use the identity:

${P}^{3} - {Q}^{3} = \left(P - Q\right) \left({P}^{2} + P Q + {Q}^{2}\right)$ ....

${\left(A - B\right)}^{3} - {B}^{3}$

$= \left(\left(A - B\right) - B\right) \left({\left(A - B\right)}^{2} + \left(A - B\right) B + {B}^{2}\right)$

$= \left(A - 2 B\right) \left({A}^{2} - 2 A B + {B}^{2} + A B - {B}^{2} + {B}^{2}\right)$

$= \left(A - 2 B\right) \left({A}^{2} - A B + {B}^{2}\right)$

This has no simpler factors with real coefficients.