Question

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

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Answer 1

Write $2z^2$ = $z^2$ + $z^2$ and then rearrange the given equation.
($z^4$ + $z^3$ + $z^2)$ +($z^2$ + z + 1) = 0
Take $z^2$ common from the first three terms of the equation $z^2$($z^2$ + z +1) + ($z^2$ + z +1) = 0
($z^2$ + z +1) ($z^2$ + 1) = 0
$z^2$ + z +1 = 0, $z^2$ + 1 = 0
z = $w,w^2,1$ $z^2$ = -1
$\omega$= $\frac{1+i\sqrt{3}}{2}$ , $w^2$= $\frac{1-i\sqrt{3}}{2}$ z = $\underset{–}{+}i$
For each value of z
$\left|z\right|$ = 1
For eg.|$\underset{–}{+}i$| = 1
$\left|w\right|=\left|w\right|=\left|\frac{1+i\sqrt{3}}{2}\right|=\frac{\sqrt{1^2+{\sqrt{3}}^2}}{2}=\frac{\sqrt{4}}{2}=\frac{2}{2}=1$ $\left|w^2\right|=\left|w^2\right|=\left|\frac{1-i\sqrt{3}}{2}\right|=\frac{\sqrt{1^2+{\sqrt{3}}^2}}{2}=\frac{\sqrt{4}}{2}=\frac{2}{2}=1$
$\left|1\right|=1$
So,we see that modulus for each of the root = 1.