Question

question
factor completely over the integers.
$x^4 - x^2 - 20$

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the polynomial becomes:
$u^2 - u - 20$

Step2: Factor the quadratic in $u$

Find two integers that multiply to $-20$ and add to $-1$: $-5$ and $4$.
$u^2 - u - 20 = (u - 5)(u + 4)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$(x^2 - 5)(x^2 + 4)$

Step4: Check for further integer factorization

$x^2 -5$ cannot be factored over integers, and $x^2 +4$ cannot be factored over integers.

Answer:

$(x^2 - 5)(x^2 + 4)$