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}.

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