Question

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

Answer 1

$z^4+z^3+2z^2+z+1=0$

$\Rightarrow (z^4+z^3+z^2)+(z^2+z+1)=0$

$\Rightarrow z^2(z^2+z+1)+1(z^2+z+1)=0$

$\Rightarrow (z^2+1)(z^2+z+1)=0$

$z^2+1=0$

$\Rightarrow z^2=-1$

We know that, $i^2=-1$

$\therefore z=i$

$\left|z\right|=\sqrt{(1{)}^2}=1$

$z^2+z+1=0$

$z=\frac{-1\pm \sqrt{1-4}}{2}$

$z=\frac{-1\pm \sqrt{3}i}{2}$

For $z=\frac{-1+\sqrt{3}i}{2}$

$\left|z\right|=\sqrt{(\frac{-1}{2}{)}^2+(\frac{\sqrt{3}}{2}{)}^2}=\sqrt{\frac{1}{4}+\frac{3}{4}}=\sqrt{\frac{4}{4}}=\sqrt{1}=1$

For $z=\frac{-1-\sqrt{3}i}{2}$

$\left|z\right|=\sqrt{(\frac{-1}{2}{)}^2+(\frac{\sqrt{3}}{2}{)}^2}=\sqrt{\frac{1}{4}+\frac{3}{4}}=\sqrt{\frac{4}{4}}=\sqrt{1}=1$

$\therefore |z|=1$