Symmetric Functions of the Roots

A symmetric function of the roots is any expression built from $\alpha_1,\dots,\alpha_n$ that is unchanged if you permute the roots among themselves — for instance $\sum\alpha^2$, $\sum\alpha^2\beta$, or $\sum \frac{1}{\alpha^2}$ for a cubic. A remarkable fact (which you do not need to prove from scratch, but should trust and use) is that every symmetric function of the roots of a polynomial equation can be written purely in terms of the coefficients $p_1,p_2,\dots$ — without ever solving the equation or knowing the individual roots. The elementary relations from the previous section ($s_1,s_2,s_3,\dots$) are the building blocks; every other symmetric function reduces to an algebraic combination of them, using ordinary identities like $(\alpha+\beta+\gamma)^2 = \sum\alpha^2 + 2\sum\alpha\beta$.

The general strategy is: start from an identity you already know connecting the quantity you want to the elementary symmetric sums, substitute in the values of $s_1,s_2,s_3,\dots$ read off from the coefficients, and simplify. This is genuinely useful — for a degree-5 or degree-6 equation you have no hope of finding the roots by hand, but you can still answer a question like 'what is the sum of the squares of the roots?' in one line.

Worked example. If $\alpha,\beta,\gamma$ are the roots of $x^3 + 3x^2 - 4x + 2 = 0$, find $\sum \alpha^2$ and $\sum \frac{1}{\alpha}$.

Here $s_1=\alpha+\beta+\gamma=-3$, $s_2=\alpha\beta+\beta\gamma+\gamma\alpha=-4$, $s_3=\alpha\beta\gamma=-2$.

For the sum of squares, use $\sum\alpha^2 = (\alpha+\beta+\gamma)^2 - 2(\alpha\beta+\beta\gamma+\gamma\alpha) = s_1^2 - 2s_2 = (-3)^2 - 2(-4) = 9+8 = 17.$ …