I was solving the polynomial $2z^3-(3-3i)z^2-(1+i)=0$ and found that they were in fact collinear!
My question is, is there a way to prove that they are collinear without explicitly finding the roots?
My thoughts so far:
- If the roots are collinear, then surely they are just multiples of each other?
- If the above is true, then we need only prove that the roots are somehow in the form $\alpha, k_1 \alpha, k_2 \alpha$ for some $k_1,k_2 \in \mathbb{R}$
- If I can show that the sum of roots (some multiple of alpha) and the sum of the roots in pairs (some multiple of alpha squared) is a multiple of the product of roots (some multiple of alpha cubed), then this is sufficient to prove the required result?
Edit to the last point above, I can see that the sum of roots in pairs is zero, so this won't work. Why is that?