If $w$ is an imaginary cube root of unity, then the polynomial whose roots are $2w+3w^2$ and $2w^2 + 3w$ is?

Question

What polynomial with complex coefficients has the following as its roots?

$2w+3w^2$ and $2w^2 + 3w$

I have tried doing this all the ways I know of, still can't get my pen over it...

Can you guys help me with this problem?

Answer 1

Assuming $w$ is a primitive third root of unity.

The sum of the roots is $5w + 5w^2 = -5$ since $1 + w + w^2 = 0$.

The product of the roots is $w^2(6 + 13w + 6w^2) = w^2(13w - 6w) = 7w^3 = 7$, using the previous fact and $w^3 = 1$.

What's a polynomial with sum of roots $-5$ and product of roots $7$?