Question

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

Answer 1

`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:
`color(white)("XXX")a^2-12a+36`
with fairly obvious factors `(a-6)^2`

Replacing the `a` with the original `x^2`
we have
`color(white)("XXX")x^4-12x^2+36=color(blue)((x^2-6)^2)`

If we regard `6` as `(sqrt(6))^2`
then `(x^2-6)` 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
`color(white)("XXX")x^4-12^x+36=[(x-sqrt(6))(x+sqrt(6))]^2`

`color(white)("XXXXXXXXXXX")=color(blue)((x-sqrt(6))^2(x+sqrt(6))^2)`