Question

question
factor completely over the integers.
$x^4 - 6x^2 + 5$

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the polynomial becomes:
$u^2 - 6u + 5$

Step2: Factor the quadratic in $u$

Find two integers that multiply to 5 and add to -6: -1 and -5.
$(u - 1)(u - 5)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$(x^2 - 1)(x^2 - 5)$

Step4: Factor difference of squares

$x^2 - 1$ is a difference of squares, so:
$(x - 1)(x + 1)(x^2 - 5)$

Answer:

$(x - 1)(x + 1)(x^2 - 5)$