If Z=x+iy is a complex number,then the number | vedclass

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(D) Given the equation $z^3+\bar{z}=0$,where $z=x+iy$ is a complex number. Taking the modulus on both sides: $|z^3| = |-\bar{z}|$. Since $|z^3| = |z|^

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$5$

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(D) Given the equation $z^3+\bar{z}=0$,where $z=x+iy$ is a complex number. Taking the modulus on both sides: $|z^3| = |-\bar{z}|$. Since $|z^3| = |z|^3$ and $|-\bar{z}| = |z|$,we have $|z|^3 = |z|$. This implies $|z|(|z|^2-1) = 0$. Case $1$: $|z|=0$,which gives the solution $z=0$. Case $2$: $|z|^2=1$,which implies $z\bar{z}=1$,so $\bar{z}=\frac{1}{z}$. Substituting $\bar{z}=\frac{1}{z}$ into the original equation: $z^3 + \frac{1}{z} = 0$,which simplifies to $z^4+1=0$. The equation $z^4 = -1$ has $4$ distinct roots. Including the solution $z=0$ from Case $1$,the total number of distinct solutions is $1 + 4 = 5$.

If $Z=x+iy$ is a complex number,then the number of distinct solutions of the equation $z^3+\bar{z}=0$ is

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