A complex number problem of equation ${z}^4 + {z}^3 + {2z}^2 + {z} + 1 = 0$

Question

If $z$ be a complex number satisfying ${z}^4 + {z}^3 + {2z}^2 + {z} + 1 = 0$, then find the value of $|\bar{z}|$

How to you proceed with this problem?

Replacing $z$ with $a + {i}b$ doesn't seem to work, as you just get a huge biquadratic equation. I do recognize that $|\bar{z}| = |z|$, but not sure how to manipulate this fact to simplify the given biquadratic equation

Answer 1

Let $w=z+\frac1z$. Then\begin{align}z^4+z^3+2z^2+z+1=0&\iff z^2+z+2+\frac1z+\frac1{z^2}=0\\&\iff\left(z+\frac1z\right)^2+z+\frac1z=0\\&\iff w^2+w=0\\&\iff w=0\vee w=-1.\end{align}So, solve the equations$$z+\frac1z=0\text{ and }z+\frac1z=-1.$$You'll see that, if $z$ is a solution, then you always get $|z|=1$. So $|\overline z|=|z|=1$.

Answer 2

Note $${z}^4 + {z}^3 + {2z}^2 + {z} + 1 =( z^4 + z^3 + z^2 )+(z^2 + z+ 1 ) \\= (z^2+1)(z^2 + z+ 1 ) =\frac{ (z^2+1)(z^3-1 ) }{z-1}=0 $$ which leads to $z^2=-1$, or $z^3=1$. As a result, $|\bar z|=1$.