Question

factor $x^4 - 2x^2 + 1$ completely.
all factors in your answer should have integer coefficients.

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, rewrite the polynomial:
$u^2 - 2u + 1$

Step2: Factor quadratic in $u$

Use perfect square trinomial rule $(a-b)^2=a^2-2ab+b^2$:
$(u - 1)^2$

Step3: Substitute back $u=x^2$

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

Step4: Factor difference of squares

Use $a^2-b^2=(a-b)(a+b)$ on $x^2-1$:
$((x - 1)(x + 1))^2$

Step5: Simplify the expression

Apply exponent to each factor:
$(x - 1)^2(x + 1)^2$

Answer:

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