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Concept
Vieta's formulas
Summary
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").
Any general polynomial of degree n
(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's 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 happens to 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 compute them.
For polynomials over a commutative ring that is not an integral domain, Vieta's formulas are only valid when is not a zero-divisor 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 set either and or and .
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 P; see .
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Related concepts (13)
Symmetric polynomial
In mathematics, a symmetric polynomial is a polynomial P(X1, X2, ..., Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2, ..., n one has P(Xσ(1), Xσ(2), ..., Xσ(n)) = P(X1, X2, ..., Xn). Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting.
Elementary symmetric polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial P is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials.
Fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.