# How do you factor (a+b)³ - (a-b)³?

May 23, 2015

Expanding:

${\left(a + b\right)}^{3} - {\left(a - b\right)}^{3}$

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

$= 6 {a}^{2} b + 2 {b}^{3}$

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

If you are allowed complex coefficients this can be broken down into linear factors:

$= 2 b \left(\sqrt{3} a + i b\right) \left(\sqrt{3} a - i b\right)$

Notice also that:

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