<p>If \( \omega \) is a non-real cube root of unity, then \( (1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 \) is:</p>
Step-by-Step Solution
Key Concept: Use 1+\omega+\omega^2=0: 1-\omega+\omega^2 = -2\omega, 1+\omega-\omega^2 = -2\omega^2. So expression = (-2\omega)^5+(-2\omega^2)^5 = -32(\omega^5+\omega^1^0) = -32(\omega^2+\omega) = -32(-1) = 32.
<p>$1+\omega+\omega^2=0 \Rightarrow 1-\omega+\omega^2 = -2\omega$, $1+\omega-\omega^2 = -2\omega^2$. Expression $ = (-2\omega)^5+(-2\omega^2)^5 = -32\omega^5-32\omega^{10} = -32\omega^2-32\omega = -32(\omega+\omega^2) = -32(-1) = 32 $.</p>
Correct Answer: A