Question

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

Explanation:

Step1: Substitute $y=x^2$

Let $y = x^2$, rewrite the polynomial:
$$y^2 - 7y + 6$$

Step2: Factor the quadratic in $y$

Find two numbers that multiply to 6 and add to -7:
$$(y - 1)(y - 6)$$

Step3: Substitute back $y=x^2$

Replace $y$ with $x^2$:
$$(x^2 - 1)(x^2 - 6)$$

Step4: Factor difference of squares

Factor $x^2-1$ using $a^2-b^2=(a-b)(a+b)$:
$$(x - 1)(x + 1)(x^2 - 6)$$

Answer:

$(x - 1)(x + 1)(x^2 - 6)$