Polar Coordinates and Complex Numbers · Challenge problem

Let $\omega=\cos72^\circ+i\sin72^\circ$. Evaluate $\omega+\omega^2+\omega^3+\omega^4$ and $\left(1+\omega\right)\left(1+\omega^2\right)\left(1+\omega^3\right)\left(1+\omega^4\right)$.