# How do you factor x^6-x^4-12x^2?

May 25, 2015

Given ${x}^{6} - {x}^{4} - 12 {x}^{2}$

Step 1: extract the obvious common factor ${x}^{2}$
$= \left({x}^{2}\right) \left({x}^{4} - {x}^{2} - 12\right)$

Step 2: simplify the second term by temporarily substituting $t = {x}^{2}$
$= \left({x}^{2}\right) \left({t}^{2} - t - 12\right)$
$= \left({x}^{2}\right) \left(t - 4\right) \left(t + 3\right)$

Step 3: re-insert ${x}^{2}$ in place of $t$
$= \left({x}^{2}\right) \left({x}^{2} - 4\right) \left({x}^{2} + 3\right)$

Step 4: factor the middle term as the difference of squares
$= \left({x}^{2}\right) \left(x + 2\right) \left(x - 2\right) \left({x}^{2} + 3\right)$