Question

question
factor completely over the integers.
$x^{4}-16x^{2}+63$

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the polynomial becomes:
$$u^2 - 16u + 63$$

Step2: Factor the quadratic in $u$

Find two integers that multiply to 63 and add to -16: -7 and -9.
$$u^2 - 16u + 63 = (u - 7)(u - 9)$$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$$(x^2 - 7)(x^2 - 9)$$

Step4: Factor difference of squares

$x^2 - 9$ is a difference of squares: $x^2 - 3^2$.
$$x^2 - 9 = (x - 3)(x + 3)$$

Answer:

$(x^2 - 7)(x - 3)(x + 3)$