Question

How do you factor `2x^3-125`?

Answer 1

`2x^3-125 = (root(3)(2)x-5)(root(3)(4)x^2+5root(3)(2)x+25)`

Explanation:

The difference of cubes identity can be written:

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

Note that `125 = 5^3` is a perfect cube (over the rationals), but `2x^3` is not.

We can treat it as a cube by using irrational coefficients, to find:

`2x^3 = (root(3)(2)x)^3`

and hence:

`2x^3-125 = (root(3)(2)x)^3-5^3`

`color(white)(2x^3-125) = (root(3)(2)x-5)((root(3)(2)x)^2+(root(3)(2)x)(5)+5^2)`

`color(white)(2x^3-125) = (root(3)(2)x-5)(root(3)(4)x^2+5root(3)(2)x+25)`

...noting that we have used `root(3)(a)root(3)(b) = root(3)(ab)` and hence:

`(root(3)(2))^2 = root(3)(2^2) = root(3)(4)`