Fundamental Theorem of Symmetric Sums

Use vieta's to get the values for the elementary symmetric sums of a polynomials' roots, then build up any symmetric polynomial of those roots in terms of those. (also newton sums)

Ex: $$x^3 + y^3 + z^3 + 2xy + 2yz + 2zx = (x + y + z)((x + y + z)^2 - 2(xy + yz + zx)) - (xy + yz + zx)((x + y + z) - 2) + 3xyz$$

In terms of symmetric sums: $$S_1(S_1^2 - 2S_2) - S_2(S_1 - 2) + 3S_3$$