If $\omega$ is a complex cube root of unity, then the value of $(1-\omega +{\omega }^2)(1+\omega -{\omega }^2)$ is equal to which of the following?

Solution

1. Given $1+\omega +{\omega }^2=0$, we have $1+{\omega }^2=-\omega$ and $1+\omega =-{\omega }^2$.
2. The expression becomes $(-\omega -\omega )(-{\omega }^2-{\omega }^2)=(-2\omega )(-2{\omega }^2)$.
3. This simplifies to $4{\omega }^3$.
4. Since ${\omega }^3=1$, the result is $4$.
5. Hence the answer is (B).