Question

How do you find all the real and complex roots of `P(z)=z^4+1`?

Answer 1

The roots of `P(z)=z^4+1` are

#z^4+1=0=>z^4=-1=>z^4=e^(2pi*i+2kpi*i)=> z=e^(pi*i/4*(1+2k))#

for `k=0,1,2,3`

So finally the roots are

`z=\pm\frac{1+i}{\sqrt 2},\ \ z=\pm\frac{1-i}{\sqrt 2}`