Question

If z is a complex number satisfying $z^4+z^3+z^2+z+1=0$, then $\left|z\right|$ is equal to ____.

Answer 1

Given that, $z^4+z^3+z^2+z+1=0$
$\Rightarrow \frac{z^5-1}{z-1}=0$ ...(sum of G.P)
$\Rightarrow z^5=1$
$\Rightarrow \left|z\right|=1$
$\text{OR}$
$z^4+z^3+z^2+z^2+z+1=0$
$\Rightarrow (z^2+(z^2+z+1)+(z^2+z+1\left)\right)=0$
$\Rightarrow (z^2+z+1)(z^2+1)=0$
$\therefore z=i,-1,\omega ,{\omega }^2.$ For each , $\left|\begin{array}{c}z\end{array}\right|=1$.