In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after (1540-1603), more commonly referred to by the Latinised form of his name, "Franciscus Vieta."
Basic formulas Any general polynomial of degree n
(with the coefficients being real or complex numbers and ) has (not necessarily distinct) complex roots by the fundamental theorem of algebra. Vieta's formulas relate the polynomial coefficients to signed sums of products of the roots as follows:
where is the th elementary symmetric polynomial in indeterminates; that is,
is the sum of all products of with different indices.
In particular,
and the above expression of degree 2 in the equals .
Generalization to rings As stated above, Vieta's formulas remain valid when the coefficients of the polynomial belong to an integral domain and the roots belong to an algebraically closed field. For a further generalization, it is useful to separate the statement in two parts: Every polynomial of degree over an integral domain factors uniquely, up to the order of the factors, as where the belong to an algebraically closed field containing . If a polynomial of degree factors over a commutative ring as then one has for , where is the th elementary symmetric polynomial, that is the sum of all products of of the with different indices.
Vieta's formulas are useful over an integral domain, because they provide relations between the roots without having to compute them. Over a ring that is not an integral domain, some care is needed. For example, in the ring of the integers modulo 8, the quadratic polynomial has four roots (1, 3, 5, and 7) and two different factorizations: As , one cannot apply Vieta's formulas with the roots and .
Example Vieta's formulas applied to quadratic and cubic polynomials:
The roots of the quadratic polynomial satisfy
The first of these equations can be used to find the minimum (or maximum) of ; see .
The roots of the cubic polynomial satisfy
Proof Vieta's formulas can be proved by considering the equality
(which is true since are all the roots of this polynomial), expanding the products in the right-hand side, and equating the coefficients of each power of between the two members of the equation.
Formally, if one expands and regroup the terms by their degree in , one gets
where the inner sum is exactly the th elementary symmetric function
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 The formulas are named after the 16th-century French mathematician (Franciscus Vieta), who derived them for the case of positive roots. However, the methods of Vieta and those of the 12th-century Islamic mathematician Sharaf al-Din al-Tusi were very close to each other.R Rashed, Résolution des équations numériques et algèbre : Saraf-al-Din al-Tusi, Viète (French), Arch. History Exact Sci. 12 (1974), 244-290. It is plausible that algebraic advancements made by other Islamic mathematicians such as Omar Khayyam, al-Tusi, and al-Kashi influenced 16th-century algebraists, with Vieta being the most prominent among them.
In the opinion of the 18th-century British mathematician Charles Hutton, as quoted by Funkhouser, the general principle (not restricted to positive real roots) was first understood by the 17th-century French mathematician Albert Girard:
See also
Content (algebra) Descartes's rule of signs Newton's identities Gauss–Lucas theorem Properties of polynomial roots Rational root theorem Symmetric polynomial and elementary symmetric polynomial
Notes
References