Question

How do you factor `4x^3 + 36`?

Answer 1

`4x^3+36 = 4(x+root(3)(9))(x^2-root(3)(9)x+3root(3)(3))`

Explanation:

First separate out the common scalar factor `4`:

`4x^3+36 = 4(x^3+9)`

Note that `x^3` is a perfect cube, but `9` is not - at least not a cube of a rational number.

We can still use the sum of cubes identity:

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

with `a=x` and `b=root(3)(9)` as follows:

`x^3+9`

`=x^3+(root(3)(9))^3`

`=(x+root(3)(9))(x^2-x(root(3)(9))+(root(3)(9))^2)`

`=(x+root(3)(9))(x^2-root(3)(9)x+root(3)(81))`

`=(x+root(3)(9))(x^2-root(3)(9)x+3root(3)(3))`

So:

`4x^3+36 = 4(x+root(3)(9))(x^2-root(3)(9)x+3root(3)(3))`