The Math Blog: Factor `x^4

Factor the number 16 and write it as a product of its prime factors:

$16 = 2 * 2 * 2 * 2$

$16 = 2^4$

So write

$2^4$ in place of

$16$ ,

$x^4 - 2^4$

Rewrite

$x^4$ as

$(x^2)^2$ and

$2^4$ as

$(2^2)^2$

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

Apply the difference of squares formula

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

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

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

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

Simplify by writing

$2^2$ as

$4$ ,

$(x^2 + 4)(x + 2)(x - 2)$

Thus you have completely factored

$x^4 - 16$ into

$(x^2 + 4)(x + 2)(x - 2)$

The Math Blog