Question

factor $x^4 - 5x^2 + 4$ completely. all factors in your answer should have integer coefficients.

Explanation:

Step1: Substitute $y=x^2$

Let $y = x^2$, so the polynomial becomes:
$y^2 - 5y + 4$

Step2: Factor quadratic in $y$

Find two integers that multiply to 4 and add to -5: -1 and -4.
$(y - 1)(y - 4)$

Step3: Substitute back $y=x^2$

Replace $y$ with $x^2$:
$(x^2 - 1)(x^2 - 4)$

Step4: Factor difference of squares

Use $a^2 - b^2=(a-b)(a+b)$ for each term:
$(x - 1)(x + 1)(x - 2)(x + 2)$

Answer:

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