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