First slide

Algebra of complex numbers

Question

If Z be a complex number satisfying $z^{4} + z^{3} + 2 z^{2} + z + 1 = 0$ then $| z |$ is equal to

Moderate

A

$\frac{1}{2}$
B

$\frac{3}{4}$
C

1
D

No unique value

Solution

The given equation is $z^{4} + z^{3} + 2 z^{2} + z + 1 = 0$
$\Rightarrow z^{4} + z^{3} + z^{2} + z^{2} + z + 1 = 0$

$\begin{matrix} \Rightarrow z^{2} (z^{2} + z + 1) + (z^{2} + z + 1) = 0 \\ \Rightarrow (z^{2} + z + 1) (z^{2} + 1) = 0 \end{matrix}$

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

$∴ z = i, - i, ω, ω^{2}, f o r e a c h | z | = 1.$

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