Question

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

Answer 1

`(x^2+16)(x+4)(x−4)`

Explanation:

`x^4 - 256`

Recall; `16^2 = 256`

`:. x^4 - 16^2`

`x^2(2) - 16^2`

Recall; `x^2 - y^2 = (x + y) (x - y) -> "difference of two squares"`

`x^2(2) - 16^2 = (x^2 + 16) (x^2 - 16)`

Factoring; `x^2 - 16 = (x + 4) (x - 4)`

Therefore;

`x^2 + 16 (x + 4) (x - 4)`

Answer 2

`(x-4)(x+4)(x+4i)(x-4i)`

Explanation:

`x^4-256" is a "color(blue)"difference of squares"`

`"which factors in general as"`

`•color(white)(x)a^2-b^2=(a-b)(a+b)`

`(x^2)^2=x^4" and "(16)^2=256`

`rArra=x^2" and "b=16`

`rArrx^4-256=(x^2-16)(x^2+16)`

`"factoring "x^2-16" as a "color(blue)"difference of squares"`

`rArrx^2-16=(x-4)(x+4)`

`"we can factor "x^2+16" by solving "x^2+16=0`

`x^2+16=0rArrx^2=-16rArrx=+-4i`

`rArrx^2+16=(x+4i)(x-4i)`

`rArrx^4-256=(x-4)(x+4)(x+4i)(x-4i)`