Using De Moivre's Theorem to Find Roots of Polynomial Equations
De Moivre's theorem states that for
where
We can obtain polynomial expressions for
and
for any
using de Moivre's theorem. For example
We can use de Moivre's theorem to find roots of some polynomial equations. Suppose we have the equation
This has the same coefficients on the right as the third and fourth equations above. We can set
then the equation becomes
Hence
Then
respectively. These are distinct, and there are no other solutions since a polynomial of degree 5 has at most 5 distinct solutions.
It is important to note that this method can only be used when the coefficients are the same as given by de Moivre's theorem for some value of n, or can be transformed into those coefficients in some way. For example, given the equation
we can substitute 2p =x and obtain the equation
solved above. We obtain as above
and use
to get