Vieta's Theorem
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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 François Viète (; 1540 – 23 February 1603), known in Latin as Franciscus Vieta, was a French people, French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as par ...
(1540-1603), more commonly referred to by the Latinised form of his name, "Franciscus Vieta."


Basic formulas

Any general polynomial of degree ''n'' P(x) = a_n x^n + a_x^ + \cdots + a_1 x + a_0 (with the coefficients being
real Real may refer to: Currencies * Argentine real * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Nature and science * Reality, the state of things as they exist, rathe ...
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 called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant polynomial, constant single-variable polynomial with Complex number, complex coefficients has at least one comp ...
. 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 \sum_ \left(\prod_^k r_\right)=(-1)^k\frac 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 a_i/a_n 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 a_n 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 r_i 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 . In other words, a field is algebraically closed if the fundamental theorem of algebra h ...
extension. Typically, is the
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
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 a_n 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 P(x) factors as a_n(x-r_1)(x-r_2)\dots(x-r_n). 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 In mathematics, a quadratic function of a single variable is a function of the form :f(x)=ax^2+bx+c,\quad a \ne 0, where is its variable, and , , and are coefficients. The expression , especially when treated as an object in itself rather tha ...
P(x) = x^2-1 has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, r_1=1 and r_2=3, because P(x)\neq (x-1)(x-3). However, P(x) does factor as (x-1)(x-7) and also as (x-3)(x-5), and Vieta's formulas hold if we set either r_1=1 and r_2=7 or r_1=3 and r_2=5.


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 r_1, r_2 of the quadratic polynomial P(x) = ax^2 + bx + c satisfy r_1 + r_2 = -\frac, \quad r_1 r_2 = \frac. The first of these equations can be used to find the minimum (or maximum) of ; see . The roots r_1, r_2, r_3 of the cubic polynomial P(x) = ax^3 + bx^2 + cx + d satisfy r_1 + r_2 + r_3 = -\frac, \quad r_1 r_2 + r_1 r_3 + r_2 r_3 = \frac, \quad r_1 r_2 r_3 = -\frac.


Proof


Direct proof

Vieta's formulas can be proved by considering the equality a_n x^n + a_x^ +\cdots + a_1 x+ a_0 = a_n (x-r_1) (x-r_2) \cdots (x-r_n) (which is true since r_1, r_2, \dots, r_n are all the roots of this polynomial), expanding the products in the right-hand side, and equating the coefficients of each power of x between the two members of the equation. Formally, if one expands (x-r_1) (x-r_2) \cdots (x-r_n) and regroup the terms by their degree in , one gets :\sum_^n (-1)^x^k \left(\sum_ r_1^\cdots r_n^\right), where the inner sum is exactly the th elementary symmetric function As an example, consider the quadratic f(x) = a_2x^2 + a_1x + a_0 = a_2(x - r_1)(x - r_2) = a_2(x^2 - x(r_1 + r_2) + r_1 r_2). Comparing identical powers of x, we find a_2=a_2, a_1=-a_2 (r_1+r_2) and a_0 = a_2 (r_1r_2) , with which we can for example identify r_1+r_2 = - a_1/a_2 and r_1r_2 = a_0/a_2 , which are Vieta's formula's for n=2.


Proof by mathematical induction

Vieta's formulas can also be proven by induction as shown below. Inductive hypothesis: Let be polynomial of degree n, with complex roots ,,, and complex coefficients a_0,a_1,\dots,a_n where \neq 0. Then the inductive hypothesis is that = ++++ = -++ Base case, n = 2 (quadratic): Let , be coefficients of the quadratic and a_0 be the constant term. Similarly, let , be the roots of the quadratic:+ + a_0 = 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 ...
:+ + a_0 = Collect
like terms In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients. Typically, in a polynomial expression, like terms are those that contain the same variables to t ...
:+ + a_0 = Apply distributive property again:+ + a_0 = The inductive hypothesis has now been proven true for n = 2. Induction step: Assuming the inductive hypothesis holds true for all n\geqslant 2, it must be true for all n+1 . = ++++By the
factor theorem In algebra, the factor theorem connects polynomial factors with polynomial roots. Specifically, if f(x) is a polynomial, then x - a is a factor of f(x) if and only if f (a) = 0 (that is, a is a root of the polynomial). The theorem is a special cas ...
, can be factored out of P(x) leaving a 0 remainder. Note that the roots of the polynomial in the square brackets are r_1,r_2,\cdots,r_n: = Factor out a_, the leading coefficient P(x), from the polynomial in the square brackets: = For simplicity sake, allow the coefficients and constant of polynomial be denoted as \zeta:P(x) = Using the inductive hypothesis, the polynomial in the square brackets can be rewritten as:P(x) = Using distributive property:P(x) = After expanding and collecting like terms:\begin = -++ \\ \endThe inductive hypothesis holds true for n+1, therefore it must be true \forall n \in \mathbb Conclusion:++++ = -++ By dividing both sides by a_, 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 Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important developments of ...
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 Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxiana) was a Persian astronomer and mathematician during the reign of Tamerlane. ...
influenced 16th-century algebraists, with Vieta being the most prominent among them. The formulas were derived by the 16th-century French mathematician
François Viète François Viète (; 1540 – 23 February 1603), known in Latin as Franciscus Vieta, was a French people, French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as par ...
, 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 an English 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 th ...
, 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 In mathematics, Descartes' rule of signs, described by René Descartes in his ''La Géométrie'', counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign chang ...
*
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 polynomi ...
*
Gauss–Lucas theorem In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometry, geometric relation between the root of a function, roots of a polynomial and the roots of its derivative . The set of roots of a real or complex polynomial ...
*
Properties of polynomial roots In mathematics, a univariate polynomial of degree with real or complex coefficients has complex ''roots'' (if counted with their multiplicities). They form a multiset of points in the complex plane, whose geometry can be deduced from the degre ...
*
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\math ...
*
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 ...


Notes


References

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