How do you factor the expression x^4 - 256?

Apr 27, 2018

(x^2+16)(x+4)(x−4)

Explanation:

${x}^{4} - 256$

Recall; ${16}^{2} = 256$

$\therefore {x}^{4} - {16}^{2}$

${x}^{2} \left(2\right) - {16}^{2}$

Recall; ${x}^{2} - {y}^{2} = \left(x + y\right) \left(x - y\right) \to \text{difference of two squares}$

${x}^{2} \left(2\right) - {16}^{2} = \left({x}^{2} + 16\right) \left({x}^{2} - 16\right)$

Factoring; ${x}^{2} - 16 = \left(x + 4\right) \left(x - 4\right)$

Therefore;

${x}^{2} + 16 \left(x + 4\right) \left(x - 4\right)$

Apr 27, 2018

$\left(x - 4\right) \left(x + 4\right) \left(x + 4 i\right) \left(x - 4 i\right)$

Explanation:

${x}^{4} - 256 \text{ is a "color(blue)"difference of squares}$

$\text{which factors in general as}$

•color(white)(x)a^2-b^2=(a-b)(a+b)

${\left({x}^{2}\right)}^{2} = {x}^{4} \text{ and } {\left(16\right)}^{2} = 256$

$\Rightarrow a = {x}^{2} \text{ and } b = 16$

$\Rightarrow {x}^{4} - 256 = \left({x}^{2} - 16\right) \left({x}^{2} + 16\right)$

$\text{factoring "x^2-16" as a "color(blue)"difference of squares}$

$\Rightarrow {x}^{2} - 16 = \left(x - 4\right) \left(x + 4\right)$

$\text{we can factor "x^2+16" by solving } {x}^{2} + 16 = 0$

${x}^{2} + 16 = 0 \Rightarrow {x}^{2} = - 16 \Rightarrow x = \pm 4 i$

$\Rightarrow {x}^{2} + 16 = \left(x + 4 i\right) \left(x - 4 i\right)$

$\Rightarrow {x}^{4} - 256 = \left(x - 4\right) \left(x + 4\right) \left(x + 4 i\right) \left(x - 4 i\right)$