Solution of an equation involving complex numbers

Question

If $z+z^3=0$ then which of the following must be true on the complex plane?
A)$Re(z)<0$
B)$Re(z)=0$
C)$Im(z) = 0$
D)$z^4=1$

My attempt:

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

Values of z satisfying the equation,
$z=0,\pm i$

So, according to me, the answer should be D), but the answer given is B)

Can anyone explain?

Answer 1

You are correct for $z=0,\pm i\,$ but note that

$$(0)^4=0$$

whereas

$$Re(0)=Re(\pm i)$$

observe indeed that in general $z=x+iy\,$ with $Re(z)=x$ and

• $0=0+i\cdot 0$
• $\pm i=0\pm i \cdot 1$

Answer 2

Since $z=0$ is one answer the options $A$ and $D$ are wrong. Also $i$ is another answer so $C$ is wrong and $B$ is true.