# How do you factor  x^4 - 12x^2 + 36?

May 11, 2016

x^4-12x^2+36 = color(blue)((x+sqrt(6))^2(x-sqrt(6))^2

#### Explanation:

If we temporarily replace ${x}^{2}$ with $a$, we would have:
$\textcolor{w h i t e}{\text{XXX}} {a}^{2} - 12 a + 36$
with fairly obvious factors ${\left(a - 6\right)}^{2}$

Replacing the $a$ with the original ${x}^{2}$
we have
$\textcolor{w h i t e}{\text{XXX}} {x}^{4} - 12 {x}^{2} + 36 = \textcolor{b l u e}{{\left({x}^{2} - 6\right)}^{2}}$

If we regard $6$ as ${\left(\sqrt{6}\right)}^{2}$
then $\left({x}^{2} - 6\right)$ can be thought of as the difference of squares with standard factors:
color(white)("XXX")(x^2-(sqrt(6)^2)=(x-sqrt(6))(x+sqrt(6))

and therefore the original equation can be further factored as
$\textcolor{w h i t e}{\text{XXX}} {x}^{4} - {12}^{x} + 36 = {\left[\left(x - \sqrt{6}\right) \left(x + \sqrt{6}\right)\right]}^{2}$

$\textcolor{w h i t e}{\text{XXXXXXXXXXX}} = \textcolor{b l u e}{{\left(x - \sqrt{6}\right)}^{2} {\left(x + \sqrt{6}\right)}^{2}}$