Smart way to check a root

Question

Let $\alpha$ be a root of $x^3+3x+5$. Let $\omega:=\frac{1+\alpha+\alpha^2}{3}$. Verify that $\omega$ is a root of $y^3+y^2+2y-1$.

Is there some trick to do this computation quickly?

Answer 1

We can write that $$ 1+\alpha+\alpha^2=\frac{1-\alpha^3}{1-\alpha} $$

But $\alpha^3 = -3\alpha-5$, so we have $$ \frac{1+\alpha+\alpha^2}3=\frac{2+\alpha}{1-\alpha} $$

It's easier to calculate from here, since you can factor out the denominator and then ignore it (because the denominator won't be zero at the root).