Algebra Examples

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

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.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Rewrite the polynomial.

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

Step 3.4

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 $x^{2}$ .

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

Step 5

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

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

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 7

Apply the product rule to $(x + 1) (x - 1)$ .

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