Build a polynomial

Question

I have $f=x^3 + ax^2 +bx +c \in \mathbb C[x], \alpha_1,\alpha_2,\alpha_3 \in \mathbb C$ are roots of $f$. $\beta_1 = {\alpha_1 \over \alpha_2} + {\alpha_2 \over \alpha_3} + {\alpha_3 \over \alpha_1}, \beta_2 = {\alpha_1 \over \alpha_3}+{\alpha_3 \over \alpha_2}+{\alpha_2 \over \alpha_1}$. I must build polynomial with $\beta_1,\beta_2$ as roots. How do I do that?

Answer 1

You should have a look to the values of $\beta_1+\beta_2$ and $\beta_1 \cdot \beta_2$ and write those quantities with elementary symmetric polynomial of $f$ roots.