Proof
Vieta's formulas can be proved by expanding the equality
(which is true since are all the roots of this polynomial), multiplying the factors on the right-hand side, and identifying the coefficients of each power of
Formally, if one expands the terms are precisely where is either 0 or 1, accordingly as whether is included in the product or not, and k is the number of that are excluded, so the total number of factors in the product is n (counting with multiplicity k) – as there are n binary choices (include or x), there are terms – geometrically, these can be understood as the vertices of a hypercube. Grouping these terms by degree yields the elementary symmetric polynomials in – for xk, all distinct k-fold products of
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