Question

question
factor completely over the integers.
$x^{4}-13x^{2}+42$

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, rewrite the polynomial:
$$u^2 - 13u + 42$$

Step2: Factor quadratic in $u$

Find two integers that multiply to 42 and add to -13: -6 and -7.
$$(u - 6)(u - 7)$$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$$(x^2 - 6)(x^2 - 7)$$

Step4: Check for further integer factorization

$x^2-6$ and $x^2-7$ cannot be factored over integers.

Answer:

$(x^2 - 6)(x^2 - 7)$