Factor x^4+2x^2+1

$x^{4} + 2 x^{2} + 1$

Step 1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} + 2 x^{2} + 1$

Step 2

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} + 2 u + 1$

Step 3

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$u^{2} + 2 u + 1^{2}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Rewrite the polynomial.

$u^{2} + 2 \cdot u \cdot 1 + 1^{2}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 1$ .

${(u + 1)}^{2}$

Step 4

Replace all occurrences of $u$ with $x^{2}$ .

${(x^{2} + 1)}^{2}$