This was a homework assignment a while back (already turned in) that I'm not totally sure how to approach. I think it might be simple but I can't wrap my mind around it. If someone could give me a tip that would be great.
I'm given a polynomial $f=x^3+ax^2+bx+c$ with roots $\alpha_1,\alpha_2,\alpha_3$. How do I find the coefficients of the following two polynomials, in terms of a,b, and c:
- the monic cubic with roots $\alpha_1^2,\alpha_2^2,\alpha_3^2$
- the monic cubic with roots $\alpha_1+\alpha_2,\alpha_1+\alpha_3,\alpha_2+\alpha_3$
What I know:
I understand that by viete's formula, $a = \alpha_1+\alpha_2+\alpha_3$, $b=\alpha_1\alpha_2+\alpha_1\alpha_3+\alpha_2\alpha_3$, and $c=\alpha_1\alpha_2\alpha_3$. But is there an elegant way to solve for the relationship between the coefficients for (1) and (2) using the coefficients of $f$? It seems resistant to a clean solution, since the relationship isn't necessarily linear.