How do I determine which complex value corresponds to which term

Question

Given that $z$ is a non real cube root of 1. Find the exact values of $a = (1+2z+3z^2)$ and $b= (1+3z+2z^2)$. I ended up getting $a+b = =-3$ and $a*b=3$. Thus solving simultaneously I conceived $z=\frac{-3±i\sqrt3}{2}$. The problem is I am not exactly sure how to determine which result is paired off with a or b. If someone could help me that would be greatly appreciated.

Answer 1

Hint:

The non-real cube roots of $1$ have minimal polynomial $1+z+z^2$. So you may write, say: $$a=1+2z+3z^2=2(\underbrace{1+z+z^2}_{=0})+z^2-1=z^2-1=\bar z-1.$$

Similarly for $b$.

Answer 2

As I read the question your obvious approach is to calculate the two values of a and the two values of b. (One for each value of z).