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

Concept: \[ 1 + \omega + \omega^2 = 0,\quad \omega^3 = 1 \]

Step 1: Use identity.
\[ \omega^2 = -1 - \omega \]

Step 2: Simplify expressions.
\[ 1 + \omega - \omega^2 = 1 + \omega + 1 + \omega = 2(1+\omega) \] \[ 1 - \omega + \omega^2 = 1 - \omega -1 - \omega = -2\omega \]

Step 3: Multiply.
\[ (2(1+\omega))(-2\omega) = -4\omega(1+\omega) \] \[ 1+\omega = -\omega^2 \Rightarrow -4\omega(-\omega^2) = 4\omega^3 = 4 \]