In mathematics, Vieta's formulas relate the coefficients of a polynomial to lớn sums and products of its roots. They are named after François Viète (more commonly referred to lớn by the Latinised size of his name, "Franciscus Vieta").
Basic formulas[edit]
Any general polynomial of degree n
(with the coefficients being real or complex numbers and a_{n} ≠ 0) has n (not necessarily distinct) complex roots r_{1}, r_{2}, ..., r_{n} by the fundamental theorem of algebra. Vieta's formulas relate the polynomial's coefficients to lớn signed sums of products of the roots r_{1}, r_{2}, ..., r_{n} as follows:
(*)

Vieta's formulas can equivalently be written as
for k = 1, 2, ..., n (the indices i_{k} are sorted in increasing order to lớn ensure each product of k roots is used exactly once).
The lefthand 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 DurandKerner method.
Generalization to lớn rings[edit]
Vieta's formulas are frequently used with polynomials with coefficients in any integral tên miền R. Then, the quotients belong to lớn the field of fractions of R (and possibly are in R itself if happens to lớn be invertible in R) and the roots are taken in an algebraically closed extension. Typically, R is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers.
Vieta's formulas are then useful because they provide relations between the roots without having to lớn compute them.
For polynomials over a commutative ring that is not an integral tên miền, Vieta's formulas are only valid when is not a zerodivisor and factors as . For example, in the ring of the integers modulo 8, the quadratic polynomial has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, and , because . However, does factor as and also as , and Vieta's formulas hold if we phối either and or and .
Example[edit]
Vieta's formulas applied to lớn quadratic and cubic polynomials:
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The roots of the quadratic polynomial satisfy
The first of these equations can be used to lớn find the minimum (or maximum) of P; see Quadratic equation § Vieta's formulas.
The roots of the cubic polynomial satisfy
Proof[edit]
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 righthand 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 included, ví 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 x^{k}, all distinct kfold products of
As an example, consider the quadratic
Comparing identical powers of , we find , and , with which we can for example identify and , which are Vieta's formula's for .
History[edit]
As reflected in the name, the formulas were discovered by the 16thcentury French mathematician François Viète, for the case of positive roots.
In the opinion of the 18thcentury British mathematician Charles Hutton, as quoted by Funkhouser,^{[1]} the general principle (not restricted to lớn positive real roots) was first understood by the 17thcentury 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.
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See also[edit]
 Content (algebra)
 Descartes' rule of signs
 Newton's identities
 Gauss–Lucas theorem
 Properties of polynomial roots
 Rational root theorem
 Symmetric polynomial and elementary symmetric polynomial
References[edit]
 "Viète theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Funkhouser, H. Gray (1930), "A short trương mục of the history of symmetric functions of roots of equations", American Mathematical Monthly, Mathematical Association of America, 37 (7): 357–365, doi:10.2307/2299273, JSTOR 2299273
 Vinberg, E. B. (2003), A course in algebra, American Mathematical Society, Providence, R.I, ISBN 0821834134
 Djukić, Dušan; et al. (2006), The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959–2004, Springer, Thành Phố New York, NY, ISBN 0387242996
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