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:

Use \(1+\omega+\omega^2=0\) to reduce everything in terms of \(\omega\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The complex cube roots of unity (\( 1, \omega, \omega^{2} \)) satisfy two fundamental identities:
1. \( 1 + \omega + \omega^{2} = 0 \).
2. \( \omega^{3} = 1 \).
Step 2: Detailed Explanation:
We need to evaluate \( (1 + \omega - \omega^{2}) (1 - \omega + \omega^{2}) \).
From \( 1 + \omega + \omega^{2} = 0 \), we can write:
\( 1 + \omega = -\omega^{2} \) and \( 1 + \omega^{2} = -\omega \).
Substitute these into the given expression:
First bracket: \( (1 + \omega) - \omega^{2} = -\omega^{2} - \omega^{2} = -2\omega^{2} \).
Second bracket: \( (1 + \omega^{2}) - \omega = -\omega - \omega = -2\omega \).
Now multiply the results:
\[ (-2\omega^{2}) \times (-2\omega) = 4\omega^{3} \]
Since \( \omega^{3} = 1 \):
\[ 4(1) = 4 \].
Step 3: Final Answer:
The value of the expression is 4.