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{{For|a method for computing {{pi}}|Viète's formula}}
In [[mathematics]], '''Vieta's formulas''' are [[formula]]s that relate the coefficients of a [[polynomial]] to sums and products of its [[Root of a function|roots]]. Named after [[François Viète]] (more commonly referred to by the Latinised form of his name, '''Franciscus Vieta'''), the formulas are used specifically in [[algebra]].


==The Laws==
===Basic formulas===


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Any general polynomial of degree ''n''
:<math>P(x)=a_nx^n + a_{n-1}x^{n-1} +\cdots + a_1 x+ a_0 \, </math>
 
(with the coefficients being real or complex numbers and ''a''<sub>''n''</sub>&nbsp;≠&nbsp;0) is known by the [[fundamental theorem of algebra]] to have ''n'' (not necessarily distinct) complex roots ''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>. Vieta's formulas relate the polynomial's coefficients {&nbsp;''a''<sub>''k''</sub>&nbsp;} to signed sums and products of its roots {&nbsp;''x''<sub>''i''</sub>&nbsp;} as follows:
 
:<math>\begin{cases} x_1 + x_2 + \dots + x_{n-1} + x_n = -\tfrac{a_{n-1}}{a_n} \\
(x_1 x_2 + x_1 x_3+\cdots + x_1x_n) + (x_2x_3+x_2x_4+\cdots + x_2x_n)+\cdots + x_{n-1}x_n = \frac{a_{n-2}}{a_n} \\
{} \quad \vdots \\ x_1 x_2 \dots x_n = (-1)^n \tfrac{a_0}{a_n}. \end{cases}</math>
 
Equivalently stated, the (''n''&nbsp;&minus;&nbsp;''k'')th coefficient ''a''<sub>''n''&minus;''k''</sub> is related to a signed sum of all possible subproducts of roots, taken ''k''-at-a-time:
 
: <math>\sum_{1\le i_1 < i_2 < \cdots < i_k\le n} x_{i_1}x_{i_2}\cdots x_{i_k}=(-1)^k\frac{a_{n-k}}{a_n}</math>
 
for ''k''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;...,&nbsp;''n'' (where we wrote the indices ''i''<sub>''k''</sub> in increasing order to ensure each subproduct of roots is used exactly once).
 
The left hand sides of Vieta's formulas are the '''[[elementary symmetric polynomial|elementary symmetric function]]s''' of the roots.
 
===Generalization to rings===
 
Vieta's formulas are frequently used with polynomials with coefficients in any [[integral domain]] ''R''. In this case the quotients <math>a_i/a_n</math> belong to the [[ring of fractions]] of ''R'' (or in ''R'' itself if <math>a_n</math> is invertible in ''R'') and the roots <math>x_i</math> are taken in an [[algebraically closed field|algebraically closed extension]]. Typically, ''R'' is the ring of the [[integer]]s, the field of fractions is the field of the [[rational number]]s and the algebraically closed field is the field of the [[complex numbers]].
 
Vieta's formulas are useful in this situation, because they provide relations between the roots without having to compute them.
 
For polynomials over a commutative ring which is not an integral domain, Vieta's formulas may be used only when the <math>a_i</math>'s are computed from the <math>x_i</math>'s. For example, in the ring of the integers [[Modular arithmetic|modulo]] 8, the polynomial <math>x^2-1</math> has four roots 1, 3, 5, 7, and Vieta's formulas are not true if, say, <math>x_1=1</math> and <math>x_2=3</math>.
 
==Example==
Vieta's formulas applied to quadratic and cubic polynomial:
 
For the [[second degree polynomial]] (quadratic) <math>P(x)=ax^2 + bx + c</math>, roots <math>x_1, x_2</math> of the equation <math>P(x)=0</math> satisfy
:<math> x_1 + x_2 = - \frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}.</math>
 
The first of these equations can be used to find the minimum (or maximum) of ''P''. See [[Quadratic equation#Vieta's formulas|second order polynomial]].
 
For the [[cubic polynomial]] <math>P(x)=ax^3 + bx^2 + cx + d</math>, roots <math>x_1, x_2, x_3</math> of the equation <math>P(x)=0</math> satisfy
:<math> x_1 + x_2 + x_3 = - \frac{b}{a}, \quad x_1 x_2 + x_1 x_3 + x_2 x_3 = \frac{c}{a}, \quad x_1 x_2 x_3 = - \frac{d}{a}.</math>
 
==Proof==
Vieta's formulas can be proved by expanding the equality
 
: <math>a_nx^n  + a_{n-1}x^{n-1} +\cdots + a_1 x+ a_0 = a_n(x-x_1)(x-x_2)\cdots (x-x_n)</math>
 
(which is true since <math>x_1, x_2, \dots, x_n</math> are all the roots of this polynomial), multiplying the factors on the right-hand side, and identifying the coefficients of each power of <math>x.</math>
 
Formally, if one expands <math>(x-x_1)(x-x_2)\cdots(x-x_n),</math> the terms are precisely <math>(-1)^{n-k}x_1^{b_1}\cdots x_n^{b_n} x^k,</math> where <math>b_i</math> is either 0 or 1, accordingly as whether <math>x_i</math> is included in the product or not, and ''k'' is the number of <math>x_i</math> that are excluded, so the total number of factors in the product is ''n'' (counting ''<math>x^k</math>'' with multiplicity ''k'') – as there are ''n'' binary choices (include <math>x_i</math> or ''x''), there are <math>2^n</math> terms – geometrically, these can be understood as the vertices of a hypercube. Grouping these terms by degree yields the elementary symmetric polynomials in <math>x_i</math> – for ''x<sup>k</sup>,'' all distinct ''k''-fold products of <math>x_i.</math>
 
== History ==
As reflected in the name, these formulas were discovered 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 in {{Harv|Funkhouser}}, the general principle (not only for positive real roots) was first understood by the 17th century French mathematician [[Albert Girard]]; Hutton writes:
<blockquote>...[Girard was] the 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.</blockquote>
 
==See also==
 
* [[Newton's identities]]
* [[Elementary symmetric polynomial]]
* [[Symmetric polynomial]]
* [[Content (algebra)]]
* [[Properties of polynomial roots]]
* [[Gauss–Lucas theorem]]
* [[Rational root theorem]]
 
== References ==
* {{springer|title=Viète theorem|id=p/v096630}}
* {{Citation| first= H. Gray | last=Funkhouser | title=A short account of the history of symmetric functions of roots of equations | journal=American Mathematical Monthly | year=1930 | volume= 37 | issue=7 | pages=357–365 | doi=10.2307/2299273| jstor= 2299273| publisher= Mathematical Association of America }}
 
*{{Citation
| last      = Vinberg
| first      = E. B.
| authorlink= Ernest Vinberg
| title      = A course in algebra
| publisher  = American Mathematical Society, Providence, R.I
| year      = 2003
| pages      =
| isbn      = 0-8218-3413-4
}}
 
*{{Citation
| last      = Djukić
| first      = Dušan, et al.
| coauthors  =
| title      = The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959–2004
| publisher  = Springer, New York, NY
| year      = 2006
| pages      =
| isbn      = 0-387-24299-6
}}
 
{{DEFAULTSORT:Viete's Formulas}}
[[Category:Articles containing proofs]]
[[Category:Polynomials]]
[[Category:Elementary algebra]]
 
{{Link GA|uz}}

Revision as of 07:52, 25 October 2013

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, Vieta's formulas are formulas that relate the coefficients of a polynomial to sums and products of its roots. Named after François Viète (more commonly referred to by the Latinised form of his name, Franciscus Vieta), the formulas are used specifically in algebra.

The Laws

Basic formulas

Any general polynomial of degree n

P(x)=anxn+an1xn1++a1x+a0

(with the coefficients being real or complex numbers and an ≠ 0) is known by the fundamental theorem of algebra to have n (not necessarily distinct) complex roots x1x2, ..., xn. Vieta's formulas relate the polynomial's coefficients { ak } to signed sums and products of its roots { xi } as follows:

{x1+x2++xn1+xn=an1an(x1x2+x1x3++x1xn)+(x2x3+x2x4++x2xn)++xn1xn=an2anx1x2xn=(1)na0an.

Equivalently stated, the (n − k)th coefficient ank is related to a signed sum of all possible subproducts of roots, taken k-at-a-time:

1i1<i2<<iknxi1xi2xik=(1)kankan

for k = 1, 2, ..., n (where we wrote the indices ik in increasing order to ensure each subproduct of roots is used exactly once).

The left hand sides of Vieta's formulas are the elementary symmetric functions of the roots.

Generalization to rings

Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R. In this case the quotients ai/an belong to the ring of fractions of R (or in R itself if an is invertible in R) and the roots xi 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 useful in this situation, because they provide relations between the roots without having to compute them.

For polynomials over a commutative ring which is not an integral domain, Vieta's formulas may be used only when the ai's are computed from the xi's. For example, in the ring of the integers modulo 8, the polynomial x21 has four roots 1, 3, 5, 7, and Vieta's formulas are not true if, say, x1=1 and x2=3.

Example

Vieta's formulas applied to quadratic and cubic polynomial:

For the second degree polynomial (quadratic) P(x)=ax2+bx+c, roots x1,x2 of the equation P(x)=0 satisfy

x1+x2=ba,x1x2=ca.

The first of these equations can be used to find the minimum (or maximum) of P. See second order polynomial.

For the cubic polynomial P(x)=ax3+bx2+cx+d, roots x1,x2,x3 of the equation P(x)=0 satisfy

x1+x2+x3=ba,x1x2+x1x3+x2x3=ca,x1x2x3=da.

Proof

Vieta's formulas can be proved by expanding the equality

anxn+an1xn1++a1x+a0=an(xx1)(xx2)(xxn)

(which is true since x1,x2,,xn are all the roots of this polynomial), multiplying the factors on the right-hand side, and identifying the coefficients of each power of x.

Formally, if one expands (xx1)(xx2)(xxn), the terms are precisely (1)nkx1b1xnbnxk, where bi is either 0 or 1, accordingly as whether xi is included in the product or not, and k is the number of xi that are excluded, so the total number of factors in the product is n (counting xk with multiplicity k) – as there are n binary choices (include xi or x), there are 2n terms – geometrically, these can be understood as the vertices of a hypercube. Grouping these terms by degree yields the elementary symmetric polynomials in xi – for xk, all distinct k-fold products of xi.

History

As reflected in the name, these formulas were discovered 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 in Template:Harv, the general principle (not only for positive real roots) was first understood by the 17th century French mathematician Albert Girard; Hutton writes:

...[Girard was] the 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

References

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