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

Question

1504 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-02-27T07:44:16

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

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

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