Question

If \(1, \omega\) and \(\omega^2\) are the cube roots of unity, then the value of \((1-\omega+\omega^2)(1+\omega-\omega^2)\) is equal to

• A. 4
• B. 0
• C. 2
• D. 3

Hint:

Always reduce everything using \(1+\omega+\omega^2=0\) — it simplifies instantly.

Answer 1

The Correct Option is A

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

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

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

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