Question

How do I factor `8x^3-1` completely?

Answer 1

Factor 8x^3 - 1

Explanation:

Algebraic identity: `a^3 - b^3 = (a - b) (a^2 + ab + b^2)`

`8x^3 - 1 = (2x - 1)(4x^2 + 2x + 1)`

Answer 2

You use the formula for the difference of perfect cubes.

Explanation:

All you really have to do in order to factor this expression completely is use the formula for the difference of two perfect cubes

`a^3 - b^3 = (a-b)(a^2 + ab + b^2)`

You can rewrite the original expression as

`8x^3 - 1 = (2x)^3 - 1^3`

This will get you

`(2x)^3 - 1^3 = (2x - 1)[(2x)^2 + 2x * 1 + 1^2]`

`(2x)^3 - 1^3 = (2x - 1) * (4x^2 + 2x + 1)`

The quadratic `4x^2 + 2x + 1` cannot be factored further without using complex numbers, which I'm not sure you're supposed to use.