# How do you factor x^3 - 4?

Apr 16, 2015

This cannot be factored using integers.

This cannot be factored using rational numbers.

This can be factored using the irrational number $\sqrt[3]{4}$

${x}^{3} - 4 = {x}^{3} - {\left(\sqrt[3]{4}\right)}^{3} = \left(x - \sqrt[3]{4}\right) \left({x}^{2} + x \sqrt[3]{4} + {\left(\sqrt[3]{4}\right)}^{2}\right)$
The quadratic above cannot be factored using real numbers.

Using Imaginary numbers we can solve: ${x}^{2} + x \sqrt[3]{4} + {\left(\sqrt[3]{4}\right)}^{2} = 0$ (Use the quadratic formula or complete the square.)

And then we can factor further:

$\left(x - \sqrt[3]{4}\right) \left(x - \left(\frac{\sqrt[3]{4}}{2} + \frac{\sqrt[3]{4} \sqrt{3}}{2} i\right)\right) \left(x - \left(\frac{\sqrt[3]{4}}{2} - \frac{\sqrt[3]{4} \sqrt{3}}{2} i\right)\right)$