Question

How do you factor `x^3 - 4`?

Answer 1

This cannot be factored using integers.

This cannot be factored using rational numbers.

This can be factored using the irrational number `root(3)4`

`x^3 - 4 = x^3 - (root(3)4) ^3 = (x - root(3)4)(x^2 + x root(3)4 + (root(3)4)^2)`
The quadratic above cannot be factored using real numbers.

Using Imaginary numbers we can solve: `x^2 + x root(3)4 + (root(3)4)^2=0` (Use the quadratic formula or complete the square.)

And then we can factor further:

`(x - root(3)4)(x - (root(3)4/2 + (root(3)4 sqrt3)/2 i))(x - (root(3)4/2 - (root(3)4 sqrt3)/2 i))`