= Solution
Use the usual convention that a <ring homomorphism> preserves the identity. Such a map from the <quotient ring> to $\mathbb C$ fixes $\mathbb Q$: it fixes integers, and preservation of inverses fixes rational numbers. It is therefore determined by the image $\zeta$ of the class of $x$. The defining relation forces $\zeta^3=1$. Conversely, evaluation at any such $\zeta$ annihilates $(x^3-1)$ and hence factors through the quotient. Thus \b[there are exactly three unital homomorphisms], namely
$$
\boxed{[f]\longmapsto f(\zeta),\qquad \zeta\in\{1,e^{2\pi i/3},e^{4\pi i/3}\}.}
$$
If the convention permits nonunital ring homomorphisms, add the zero map. These are the only extra possibilities: the image of $1$ is an idempotent in $\mathbb C$, hence either zero or one, and image zero forces the whole map to vanish.
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