Vieta's formulas

Vieta's formulas are the general principle behind “sum and product of roots”, extended from quadratics to polynomials of any degree. They reveal that the coefficients of a polynomial are nothing more than symmetric combinations of its roots — sums, sums of pairwise products, and so on. Named after the French mathematician François Viète.

For a quadratic x^2 + bx + c with roots r and s: r + s = -b and r·s = c. For a cubic x^3 + bx^2 + cx + d with roots r, s, t: r + s + t = -b, rs + rt + st = c, and rst = -d. The pattern: each coefficient (with alternating signs) equals a particular symmetric sum of the roots.

These formulas are powerful because they go backward — from roots to coefficients — and they hold even when the roots are irrational or complex. They are a staple of competition mathematics and a stepping stone toward Galois theory, where the deep relationship between a polynomial's roots and its coefficients becomes the central object of study.