Question

If $\omega$ is a cube root of unity, then $(1 + \omega - \omega^{2})(1 - \omega + \omega^{2})$ is

• A. 1
• B. 0
• C. 2
• D. 4

Hint:

Key properties of cube roots of unity: $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$. Rewrite expressions using the second identity to simplify quickly.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Use $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.
Step 2: Detailed Explanation:
$1+\omega-\omega^2 = (1+\omega+\omega^2)-2\omega^2 = -2\omega^2$.
$1-\omega+\omega^2 = (1+\omega+\omega^2)-2\omega = -2\omega$.
Product $= (-2\omega^2)(-2\omega) = 4\omega^3 = 4(1) = 4$.
Step 3: Final Answer:
The product equals $4$.