Using De Moivre's Theorem to Find Roots of Polynomial Equations

De Moivre's theorem states that forwhere

We can obtain polynomial expressions forandfor anyusing de Moivre's theorem. For example

We can use de Moivre's theorem to find roots of some polynomial equations. Suppose we have the equationThis has the same coefficients on the right as the third and fourth equations above. We can setthen the equation becomes

Hence

Thenrespectively. 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 equationwe can substitute 2p =x and obtain the equationsolved above. We obtain as above

and useto get