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The question appears to be:

"If $\omega$ is a cube root of unity, then $1 + \omega + \omega^2 =$
(a) 1
(b) 2
(c) 3
(d) 0"

This is a common problem involving cube roots of unity.

Solution:

The cube roots of unity are the solutions to the equation $x^3 = 1$, which are $1$, $\omega$, and $\omega^2$, where $\omega = e^{2\pi i / 3}$ and $\omega^2 = e^{4\pi i / 3}$. The properties of $\omega$ include:

1. $\omega^3 = 1$
2. $1 + \omega + \omega^2 = 0$

Therefore, the correct answer is $1 + \omega + \omega^2 = 0$, which corresponds to option (d) 0.

Would you like further details or have any questions about this solution?

Related Questions:

1. What are the geometric interpretations of the cube roots of unity?
2. How are cube roots of unity used in polynomial factorization?
3. Can you explain the significance of roots of unity in number theory?
4. How does the equation change if we consider $n$th roots of unity instead of cube roots?
5. How would the problem differ if $\omega$ was a square root of unity?

Tip: Roots of unity are fundamental in understanding cyclic groups and symmetry operations in mathematics.