## Compute some symmetric combinations of the roots of a given polynomial

Let $\alpha$ and $\beta$ be the roots of $p(x) = x^2-5x+7$. Compute $\alpha+\beta$, $\alpha\beta$, $\alpha^2+\beta^2$, and $\alpha^3+\beta^3$.

We can readily see that, without loss of generality, $\alpha = \frac{5+3i}{2}$ and $\beta = \frac{5-3i}{2}$.

Let $\sigma_1 = x_1+x_2$ and $\sigma_2 = x_1x_2$ be the elementary symmetric polynomials in two variables. Our task is to evaluate the (symmetric) polynomials $x_1+x_2$, $x_1x_2$, $x_1^2+x_2^2$, and $x_1^3+x_2^3$ at $(\alpha,\beta)$. Note that these are precisely $\sigma_1$, $\sigma_2$, $\sigma_1^2-\sigma_2$, and $\sigma_1^3 - 3\sigma_1\sigma_2$.

We can directly compute $\alpha+\beta = 5$ and $\alpha\beta = \frac{17}{2}$ easily enough. Now $\alpha^2 + \beta^2 = \frac{33}{2}$ and $\alpha^3 + \beta^3 = \frac{-5}{2}$.

Note that once $\sigma_1$ and $\sigma_2$ are computed, all other symmetric combinations of $\alpha$ and $\beta$ can be computed entirely inside $\mathbb{Q}$.