First slide

Algebra of complex numbers

Question

If $i z^{3} + z^{2} - z + i = 0$ , then

Moderate

A

$Re z = 0$
B

$Im z = 0$
C

$| z | = 1$
D

none of these

Solution

$\begin{array}{l} i z^{3} + z^{2} - z + i = 0 \Rightarrow i z^{3} - i^{2} z^{2} - (z - i) = 0 \\ \Rightarrow i z^{2} (z - i) - (z - i) = 0 \\ \Rightarrow (i z^{2} - 1) (z - i) = 0 \\ \Rightarrow z^{2} = 1 / i or z = i \end{array}$
In any case $| z | = 1$

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