Question

factor ( x^4 - 8x^2 + 16 ) completely. all factors in your answer should have integer coefficients.

Explanation:

Step1: Substitute $y=x^2$

Let $y = x^2$, so the expression becomes:
$y^2 - 8y + 16$

Step2: Factor quadratic in $y$

This is a perfect square trinomial:
$y^2 - 8y + 16 = (y - 4)^2$

Step3: Substitute back $y=x^2$

Replace $y$ with $x^2$:
$(x^2 - 4)^2$

Step4: Factor difference of squares

$x^2 - 4$ is a difference of squares, so:
$x^2 - 4 = (x - 2)(x + 2)$
Substitute back to get the fully factored form:
$((x - 2)(x + 2))^2 = (x - 2)^2(x + 2)^2$

Answer:

$(x - 2)^2(x + 2)^2$