Question

If \( 1, \omega, \omega^2 \) are the cube roots of unity and \( f(x, y) = (x + y)(x\omega + y\omega^2)(x\omega^2 + y\omega) \), then \( f(2, 3) \) is:

• A. \(16\)
• B. \(24\)
• C. \(35\)
• D. \(45\)

Hint:

Use the identity \( 1 + \omega + \omega^2 = 0 \) and \( \omega^3 = 1 \) to simplify expressions involving cube roots of unity.

Answer 1

The Correct Option is C

\[ f(2, 3) = (2 + 3)(2\omega + 3\omega^2)(2\omega^2 + 3\omega) = 5(2\omega + 3\omega^2)(2\omega^2 + 3\omega) \] Now simplify: \[ \begin{aligned} &= 5\left[(2\omega)(2\omega^2) + (2\omega)(3\omega) + (3\omega^2)(2\omega^2) + (3\omega^2)(3\omega)\right]
&= 5\left[4 + 6\omega^2 + 6\omega + 9\right]
&= 5\left(13 + 6(\omega + \omega^2)\right)
&= 5(13 - 6) \quad \text{(since } \omega + \omega^2 = -1\text{)}
&= 35 \end{aligned} \]