Question

solve the following using demoiver's theorem

x^5-1=0

Answer 1

as x5+1=0, we have x5=−1 and x=5√−1=(−1)15

Hence solution of x5+1=0 means to find fifth roots of −1.

Note that as −1=cosπ+isinπ, and we can also write

−1=cos(2nπ+π)+isin(2nπ+π)

and using De Moivre's Theorem

(−1)15=cos(2nπ+π5)+isin(2nπ+π5)

and five roots, which are solutions of x5+1=0 can be obtained by putting n=0,1,2,3 and 4 (after 4 roots will start repeating) and these are

cos(π5)+isin(π5)
cos(3π5)+isin(3π5)=−cos(2π5)+isin(2π5)
cos(5π5)+isin(5π5)=cosπ+isinπ=−1
cos(7π5)+isin(7π5)=−cos(2π5)−isin(2π5)
cos(9π5)+isin(9π5)=cos(π5)−isin(π5)