Question

The number of solutions for $z^3+\overline{z}=0$ is

• A. 4
  
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is A 4

$z^3+\overline{z}=0$ ...(1)

$\Rightarrow z^3=-\overline{z}$

$\Rightarrow |z{|}^3=|(-\overline{z})|$

$\Rightarrow |z{|}^3=|z|\Rightarrow |z{|}^3-|z|=0$

$\Rightarrow |z|(|z{|}^2-1)=0$

$\Rightarrow |z|=0,|z{|}^2=1$

$\Rightarrow |z|=\pm 1$

$|z{|}^2=z\cdot \overline{z}=1$

From equation (1), we get

$\Rightarrow z^3+\frac{1}{z}=0$

$\Rightarrow z^4+1=0$

So, the equation has four roots.