Vieta's formulas

In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

[1] They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").

(with the coefficients being real or complex numbers and an ≠ 0) has n (not necessarily distinct) complex roots r1, r2, ..., rn by the fundamental theorem of algebra.

Vieta's formulas relate the polynomial coefficients to signed sums of products of the roots r1, r2, ..., rn as follows: Vieta's formulas can equivalently be written as

for k = 1, 2, ..., n (the indices ik are sorted in increasing order to ensure each product of k roots is used exactly once).

The left-hand sides of Vieta's formulas are the elementary symmetric polynomials of the roots.

Vieta's system (*) can be solved by Newton's method through an explicit simple iterative formula, the Durand-Kerner method.

Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R. Then, the quotients

belong to the field of fractions of R (and possibly are in R itself if

Vieta's formulas are then useful because they provide relations between the roots without having to compute them.

For polynomials over a commutative ring that is not an integral domain, Vieta's formulas are only valid when

For example, in the ring of the integers modulo 8, the quadratic polynomial

Vieta's formulas applied to quadratic and cubic polynomials: The roots

are all the roots of this polynomial), expanding the products in the right-hand side, and equating the coefficients of each power of

that are included, so the total number of factors in the product is n (counting

with multiplicity k) – as there are n binary choices (include

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

Vieta's formulas can also be proven by induction as shown below.

Expand the right side using distributive property:

The inductive hypothesis has now been proven true for

Note that the roots of the polynomial in the square brackets are

For simplicity sake, allow the coefficients and constant of polynomial be denoted as

Using the inductive hypothesis, the polynomial in the square brackets can be rewritten as:

A method similar to Vieta's formula can be found in the work of the 12th century Arabic mathematician Sharaf al-Din al-Tusi.

It is plausible that the algebraic advancements made by Arabic mathematicians such as al-Khayyam, al-Tusi, and al-Kashi influenced 16th-century algebraists, with Vieta being the most prominent among them.

[2][3] The formulas were derived 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 by Funkhouser,[4] the general principle (not restricted to positive real roots) was first understood by the 17th-century French mathematician Albert Girard: ...[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.