Question

How do you factor completely: `16x^4 - 1`?

Answer 1

`16x^4-1=color(blue)((4x^2+1)(2x+1)(2x-1)`

Explanation:

Remember
`color(white)("XXX")(a^2-b^2)=(a+b)(a-b)`

`(16x^4-1)` is of this form (with `a=(4x^2)` and `b=1`).

So
`color(white)("XXX")(16x^4-1)=(4x^2+1)(4x^2-1)`

...but we note that `(4x^2-1)` is also of this form (with `a=2x` and `b=1`)
So `(4x^2-1)=(2x+1)(2x-1)`

and, completely factored we have
`color(white)("XXX")(16x^4-1)=(4x^2+1)(2x+1)(2x-1)`