The Math Blog: Factor `x^4

Rewrite 1 as

$1^4$ (one raised to the power of four),

$x^4 - 1^4$

You can write

$x^4$ as

$(x^2)^2$ and

$1^4$ as

$(1^2)^2$ ,

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

Apply the difference of squares formula

$a^2 - b^2 = (a + b)(a - b)$ ,

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

Again apply the difference of squares formula on the second parenthesis above,

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

Simplify by rewriting

$1^2$ as 1,

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

Thus you have completely factored the expression

$x^4 - 1$ into

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

The Math Blog