In mathematics, Vieta's formulas relate the
coefficients of a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
to sums and products of its
roots. They are named after
François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
(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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
numbers and ) has (not necessarily distinct) complex roots by the
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 polynomia ...
. Vieta's formulas relate the polynomial's coefficients to signed sums of products of the roots as follows:
:
Vieta's formulas can equivalently be written as
:
for (the indices are sorted in increasing order to ensure each product of roots is used exactly once).
The left-hand sides of Vieta's formulas are the
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 sy ...
s of the roots.
Generalization to rings
Vieta's formulas are frequently used with polynomials with coefficients in any
integral domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
. Then, the quotients
belong to the
field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of (and possibly are in itself if
happens to be
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
in ) and the roots
are taken in an
algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
extension. Typically, is the
ring of the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, the field of fractions is the
field of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s and the algebraically closed field is the field of the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s.
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
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a 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
.
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 expanding the equality
:
(which is true since
are all the roots of this polynomial), multiplying the factors on the right-hand 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, so 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 ''k''-fold 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
As reflected in the name, the formulas were discovered by the 16th-century French mathematician
François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
, for the case of positive roots.
In the opinion of the 18th-century British mathematician
Charles Hutton
Charles Hutton FRS FRSE LLD (14 August 1737 – 27 January 1823) was a British mathematician and surveyor. He was professor of mathematics at the Royal Military Academy, Woolwich from 1773 to 1807. He is remembered for his calculation of the ...
, as quoted by Funkhouser,
the general principle (not restricted to positive real roots) was first understood by the 17th-century French mathematician
Albert Girard:
... irard wasthe 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.
See also
*
Content (algebra)
In algebra, the content of a polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primitive part of such a polynomial is the quotient ...
*
Descartes' rule of signs
In mathematics, Descartes' rule of signs, first described by René Descartes in his work ''La Géométrie'', is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots i ...
*
Newton's identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynom ...
*
Gauss–Lucas theorem
In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial ''P'' and the roots of its derivative ''P′''. The set of roots of a real or complex polynomial is a set of points ...
*
Properties of polynomial roots
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy an ...
*
Rational root theorem
In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or theorem) states a constraint on rational solutions of a polynomial equation
:a_nx^n+a_x^+\cdots+a_0 = 0
with integer coefficients a_i\in ...
*
Symmetric polynomial
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has ...
and
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 sy ...
References
*
*
*
*
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Articles containing proofs
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Elementary algebra