# How do I use DeMoivre's theorem to solve #z^3-1=0#?

##### 2 Answers

If

If you're using complex numbers, then every polynomial equation of degree

De Moivre's theorem uses the fact that we can write any complex number as

If you look at

Since

This means that the three solutions are:

#\rho=1, \theta=0# , which is the real number#1# .#\rho=1, \theta=\frac{2\pi}{3}# , which is the complex number#-1/2 + \sqrt{3}/2 i# #\rho=1, \theta=\frac{4\pi}{3}# , which is the complex number#-1/2 - \sqrt{3}/2 i#

#### Explanation:

We know that any complex number,

So

Use de Moivre's theorem:

Now we must consider every k such that

These values are called the cubic roots of unity and are usually written as

The fact that