Question

How do you factor `1/8 + 8x^3`?

Answer 1

Use the sum of cubes identity to find:

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

Explanation:

The sum of cubes identity can be written:

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

Both `1/8 = (1/2)^3` and `8x^3 = (2x)^3` are perfect cubes, so it is natural to use the sum of cubes identity to help factor this, putting `a=1/2` and `b=2x` ...

`1/8+8x^3`

`=(1/2)^3+(2x)^3`

`=(1/2+2x)((1/2)^2-(1/2)(2x)+(2x)^2)`

`=(1/2+2x)(1/4-x+4x^2)`