Question

question
factor completely over the integers.
$x^4 - 6x^2 - 7$

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the polynomial becomes:
$$u^2 - 6u - 7$$

Step2: Factor the quadratic in $u$

Find two integers that multiply to $-7$ and add to $-6$: $-7$ and $1$.
$$u^2 - 6u - 7 = (u - 7)(u + 1)$$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$$(x^2 - 7)(x^2 + 1)$$

Step4: Check for further integer factors

$x^2 -7$ cannot be factored over integers, and $x^2 +1$ also cannot be factored over integers.

Answer:

$(x^2 + 1)(x^2 - 7)$