
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Vieta's formulas relate the
coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s of a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
to sums and products of its
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusin ...
. They are named after
François Viète (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
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. Vieta's formulas relate the polynomial 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.
Vieta's system can be solved by
Newton's method
In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a ...
through an explicit simple iterative formula, the
Durand-Kerner method.
Generalization to rings
Vieta's formulas are frequently used with polynomials with coefficients in any
integral domain
In mathematics, 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 setting for studying divisibilit ...
. 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 fie ...
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 ...
in ) and the roots
are taken in an
algebraically closed extension. Typically, is the
ring of the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 (for example,
The set of all ...
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 for ...
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
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
that is not an integral domain, Vieta's formulas are only valid when
is not a
zero-divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zer ...
and
factors as
. For example, in the ring of the integers
modulo
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation.
Given two positive numbers and , mo ...
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
.
Example
Vieta's formulas applied to
quadratic and
cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
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
Direct 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
.
Proof by mathematical induction
Vieta's formulas can also be proven by
induction as shown below.
Inductive hypothesis:
Let
be polynomial of degree
, with complex roots
and complex coefficients
where
. Then the inductive hypothesis is that
Base case,
(quadratic):
Let
be coefficients of the quadratic and
be the constant term. Similarly, let
be the roots of the quadratic:
Expand the right side using
distributive property
In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary ...
:
Collect
like terms:
Apply distributive property again:
The inductive hypothesis has now been proven true for
.
Induction step:
Assuming the inductive hypothesis holds true for all
, it must be true for all
.
By the
factor theorem,
can be factored out of
leaving a 0 remainder. Note that the roots of the polynomial in the square brackets are
:
Factor out
, the leading coefficient
, from the polynomial in the square brackets:
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:
Using distributive property:
After expanding and collecting like terms:
The inductive hypothesis holds true for
, therefore it must be true
Conclusion:
By dividing both sides by
, it proves the Vieta's formulas true.
History
A method similar to Vieta's formula can be found in the work of the 12th century
Islamic mathematician Sharaf al-Din al-Tusi. It is plausible that algebraic advancements made by other Islamic mathematicians such as
Omar Khayyam
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīshābūrī (18 May 1048 – 4 December 1131) (Persian language, Persian: غیاث الدین ابوالفتح عمر بن ابراهیم خیام نیشابورﻯ), commonly known as Omar ...
,
al-tusi, and
al-Kashi influenced 16th-century algebraists, with Vieta being the most prominent among them.
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,
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 nonzero 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 ...
*
Descartes' rule of signs
*
Newton's identities
*
Gauss–Lucas theorem
*
Properties of polynomial roots
*
Rational root theorem
*
Symmetric polynomial 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 ...
Notes
References
*
*
*
*
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