Question

How do you find all the real and complex roots of `F(x)=x^5-1`?

Answer 1

See below

Explanation:

`x^5-1=0` or

`x=1e^{i phi/5}` where `phi=arctan0+2kpi,k=pm1,pm2,pm3,cdots`

then

`x_k=e^{i(2k pi)/5}` for `k=0,1,2,3,4` are the roots or

`{(x = 1), (x = -(1+sqrt[5])/4 - i sqrt[(5- sqrt[5])/8]), (x= -(1- sqrt[5])/4 + i sqrt[(5+ sqrt[5])/8]), (x= -(1- sqrt[5])/4 - i sqrt[(5 + sqrt[5])/8]), (x = -(1+sqrt[5])/4 + i sqrt[(5 - sqrt[5])/8]):}`

of course we used the Moivre's identity

`e^{iphi} = cos(phi)+isin(phi)`