In mathematics, Vieta's formulas are formulas that relate the coefficients of a polynomial to sums and products of its roots. Named after François Viète (more commonly referred to by the Latinised form of his name, Franciscus Vieta), the formulas are used specifically in algebra.
Contents
Basic formulas
Any general polynomial of degree n
(with the coefficients being real or complex numbers and an ≠ 0) is known by the fundamental theorem of algebra to have n (not necessarily distinct) complex roots x1, x2, ..., xn. Vieta's formulas relate the polynomial's coefficients { ak } to signed sums and products of its roots { xi } as follows:
Equivalently stated, the (n − k)th coefficient an−k is related to a signed sum of all possible subproducts of roots, taken k-at-a-time:
for k = 1, 2, ..., n (where we wrote the indices ik in increasing order to ensure each subproduct of roots is used exactly once).
The left hand sides of Vieta's formulas are the elementary symmetric functions of the roots.
Generalization to rings
Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R. In this case the quotients
Vieta's formulas are useful in this situation, because they provide relations between the roots without having to compute them.
For polynomials over a commutative ring which is not an integral domain, Vieta's formulas are only valid when
Example
Vieta's formulas applied to quadratic and cubic polynomial:
For the second degree polynomial (quadratic)
The first of these equations can be used to find the minimum (or maximum) of P. See second order polynomial.
For the cubic polynomial
Proof
Vieta's formulas can be proved by expanding the equality
(which is true since
Formally, if one expands
History
As reflected in the name, these formulas were discovered by the 16th century French mathematician François Viète, for the case of positive roots.
In the opinion of the 18th century British mathematician Charles Hutton, as quoted in (Funkhouser), the general principle (not only for positive real roots) was first understood by the 17th century French mathematician Albert Girard; Hutton writes:
...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.