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").
(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 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).
Vieta's formulas are frequently used with polynomials with coefficients in any integral domainR. 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 .
Exampleedit
Vieta's formulas applied to quadratic and cubic polynomials:
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 xk, 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 .
Alternate proof (mathematical induction)edit
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:
The inductive hypothesis has now been proven true for n = 2.
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 both sides by , it proves the Vieta's formulas true.
Historyedit
As reflected in the name, the 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 by Funkhouser,[1] the general principle (not restricted to positive real roots) was first understood by the 17th-century French mathematician Albert Girard:
...[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.
Funkhouser, H. Gray (1930), "A short account of the history of symmetric functions of roots of equations", American Mathematical Monthly, 37 (7), Mathematical Association of America: 357–365, doi:10.2307/2299273, JSTOR 2299273
Djukić, Dušan; et al. (2006), The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959–2004, Springer, New York, NY, ISBN0-387-24299-6
April 08, 2024
vieta, formulas, method, computing, viète, formula, mathematics, relate, coefficients, polynomial, sums, products, roots, they, named, after, françois, viète, more, commonly, referred, latinised, form, name, franciscus, vieta, françois, viète, contents, basic,. For a method for computing p see Viete s formula In mathematics Vieta s formulas relate the coefficients of a polynomial to sums and products of its roots They are named after Francois Viete more commonly referred to by the Latinised form of his name Franciscus Vieta Francois Viete Contents 1 Basic formulas 2 Generalization to rings 3 Example 4 Proof 4 1 Alternate proof mathematical induction 5 History 6 See also 7 ReferencesBasic formulas editAny general polynomial of degree nP x anxn an 1xn 1 a1x a0 displaystyle P x a n x n a n 1 x n 1 cdots a 1 x a 0 nbsp 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 coefficients to signed sums of products of the roots r1 r2 rn as follows r1 r2 rn 1 rn an 1an r1r2 r1r3 r1rn r2r3 r2r4 r2rn rn 1rn an 2an r1r2 rn 1 na0an displaystyle begin cases r 1 r 2 dots r n 1 r n dfrac a n 1 a n 1ex r 1 r 2 r 1 r 3 cdots r 1 r n r 2 r 3 r 2 r 4 cdots r 2 r n cdots r n 1 r n dfrac a n 2 a n 1ex quad vdots 1ex r 1 r 2 cdots r n 1 n dfrac a 0 a n end cases nbsp Vieta s formulas can equivalently be written as 1 i1 lt i2 lt lt ik n j 1krij 1 kan kan displaystyle sum 1 leq i 1 lt i 2 lt cdots lt i k leq n left prod j 1 k r i j right 1 k frac a n k a n nbsp 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 Generalization to rings editVieta s formulas are frequently used with polynomials with coefficients in any integral domain R Then the quotients ai an displaystyle a i a n nbsp belong to the field of fractions of R and possibly are in R itself if an displaystyle a n nbsp happens to be invertible in R and the roots ri displaystyle r i nbsp 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 an displaystyle a n nbsp is not a zero divisor and P x displaystyle P x nbsp factors as an x r1 x r2 x rn displaystyle a n x r 1 x r 2 dots x r n nbsp For example in the ring of the integers modulo 8 the quadratic polynomial P x x2 1 displaystyle P x x 2 1 nbsp has four roots 1 3 5 and 7 Vieta s formulas are not true if say r1 1 displaystyle r 1 1 nbsp and r2 3 displaystyle r 2 3 nbsp because P x x 1 x 3 displaystyle P x neq x 1 x 3 nbsp However P x displaystyle P x nbsp does factor as x 1 x 7 displaystyle x 1 x 7 nbsp and also as x 3 x 5 displaystyle x 3 x 5 nbsp and Vieta s formulas hold if we set either r1 1 displaystyle r 1 1 nbsp and r2 7 displaystyle r 2 7 nbsp or r1 3 displaystyle r 1 3 nbsp and r2 5 displaystyle r 2 5 nbsp Example editVieta s formulas applied to quadratic and cubic polynomials The roots r1 r2 displaystyle r 1 r 2 nbsp of the quadratic polynomial P x ax2 bx c displaystyle P x ax 2 bx c nbsp satisfyr1 r2 ba r1r2 ca displaystyle r 1 r 2 frac b a quad r 1 r 2 frac c a nbsp The first of these equations can be used to find the minimum or maximum of P see Quadratic equation Vieta s formulas The roots r1 r2 r3 displaystyle r 1 r 2 r 3 nbsp of the cubic polynomial P x ax3 bx2 cx d displaystyle P x ax 3 bx 2 cx d nbsp satisfyr1 r2 r3 ba r1r2 r1r3 r2r3 ca r1r2r3 da displaystyle r 1 r 2 r 3 frac b a quad r 1 r 2 r 1 r 3 r 2 r 3 frac c a quad r 1 r 2 r 3 frac d a nbsp Proof editVieta s formulas can be proved by expanding the equalityanxn an 1xn 1 a1x a0 an x r1 x r2 x rn displaystyle a n x n a n 1 x n 1 cdots a 1 x a 0 a n x r 1 x r 2 cdots x r n nbsp which is true since r1 r2 rn displaystyle r 1 r 2 dots r n nbsp are all the roots of this polynomial multiplying the factors on the right hand side and identifying the coefficients of each power of x displaystyle x nbsp Formally if one expands x r1 x r2 x rn displaystyle x r 1 x r 2 cdots x r n nbsp the terms are precisely 1 n kr1b1 rnbnxk displaystyle 1 n k r 1 b 1 cdots r n b n x k nbsp where bi displaystyle b i nbsp is either 0 or 1 accordingly as whether ri displaystyle r i nbsp is included in the product or not and k is the number of ri displaystyle r i nbsp that are included so the total number of factors in the product is n counting xk displaystyle x k nbsp with multiplicity k as there are n binary choices include ri displaystyle r i nbsp or x there are 2n displaystyle 2 n nbsp terms geometrically these can be understood as the vertices of a hypercube Grouping these terms by degree yields the elementary symmetric polynomials in ri displaystyle r i nbsp for xk all distinct k fold products of ri displaystyle r i nbsp As an example consider the quadraticf x a2x2 a1x a0 a2 x r1 x r2 a2 x2 x r1 r2 r1r2 displaystyle f x a 2 x 2 a 1 x a 0 a 2 x r 1 x r 2 a 2 x 2 x r 1 r 2 r 1 r 2 nbsp Comparing identical powers of x displaystyle x nbsp we find a2 a2 displaystyle a 2 a 2 nbsp a1 a2 r1 r2 displaystyle a 1 a 2 r 1 r 2 nbsp and a0 a2 r1r2 displaystyle a 0 a 2 r 1 r 2 nbsp with which we can for example identify r1 r2 a1 a2 displaystyle r 1 r 2 a 1 a 2 nbsp and r1r2 a0 a2 displaystyle r 1 r 2 a 0 a 2 nbsp which are Vieta s formula s for n 2 displaystyle n 2 nbsp Alternate proof mathematical induction edit Vieta s formulas can also be proven by induction as shown below Inductive hypothesis Let P x displaystyle P x nbsp be polynomial of degree n displaystyle n nbsp with complex roots r1 r2 rn displaystyle r 1 r 2 dots r n nbsp and complex coefficients a0 a1 an displaystyle a 0 a 1 dots a n nbsp where an 0 displaystyle a n neq 0 nbsp Then the inductive hypothesis is thatP x anxn an 1xn 1 a1x a0 anxn an r1 r2 rn xn 1 1 n an r1r2 rn displaystyle P x a n x n a n 1 x n 1 cdots a 1 x a 0 a n x n a n r 1 r 2 cdots r n x n 1 cdots 1 n a n r 1 r 2 cdots r n nbsp Base case n 2 displaystyle n 2 nbsp quadratic Let a2 a1 displaystyle a 2 a 1 nbsp be coefficients of the quadratic and a0 displaystyle a 0 nbsp be the constant term Similarly let r1 r2 displaystyle r 1 r 2 nbsp be the roots of the quadratic a2x2 a1x a0 a2 x r1 x r2 displaystyle a 2 x 2 a 1 x a 0 a 2 x r 1 x r 2 nbsp Expand the right side using distributive property a2x2 a1x a0 a2 x2 r1x r2x r1r2 displaystyle a 2 x 2 a 1 x a 0 a 2 x 2 r 1 x r 2 x r 1 r 2 nbsp Collect like terms a2x2 a1x a0 a2 x2 r1 r2 x r1r2 displaystyle a 2 x 2 a 1 x a 0 a 2 x 2 r 1 r 2 x r 1 r 2 nbsp Apply distributive property again a2x2 a1x a0 a2x2 a2 r1 r2 x a2 r1r2 displaystyle a 2 x 2 a 1 x a 0 a 2 x 2 a 2 r 1 r 2 x a 2 r 1 r 2 nbsp The inductive hypothesis has now been proven true for n 2 Induction step Assuming the inductive hypothesis holds true for all n 2 displaystyle n geqslant 2 nbsp it must be true for all n 1 displaystyle n 1 nbsp P x an 1xn 1 anxn a1x a0 displaystyle P x a n 1 x n 1 a n x n cdots a 1 x a 0 nbsp By the factor theorem x rn 1 displaystyle x r n 1 nbsp can be factored out of P x displaystyle P x nbsp leaving a 0 remainder Note that the roots of the polynomial in the square brackets are r1 r2 rn displaystyle r 1 r 2 cdots r n nbsp P x x rn 1 an 1xn 1 anxn a1x a0x rn 1 displaystyle P x x r n 1 frac a n 1 x n 1 a n x n cdots a 1 x a 0 x r n 1 nbsp Factor out an 1 displaystyle a n 1 nbsp the leading coefficient P x displaystyle P x nbsp from the polynomial in the square brackets P x an 1 x rn 1 xn 1 anxn an 1 a1 an 1 x a0 an 1 x rn 1 displaystyle P x a n 1 x r n 1 frac x n 1 frac a n x n a n 1 cdots frac a 1 a n 1 x frac a 0 a n 1 x r n 1 nbsp For simplicity sake allow the coefficients and constant of polynomial be denoted as z displaystyle zeta nbsp P x an 1 x rn 1 xn zn 1xn 1 z0 displaystyle P x a n 1 x r n 1 x n zeta n 1 x n 1 cdots zeta 0 nbsp Using the inductive hypothesis the polynomial in the square brackets can be rewritten as P x an 1 x rn 1 xn r1 r2 rn xn 1 1 n r1r2 rn displaystyle P x a n 1 x r n 1 x n r 1 r 2 cdots r n x n 1 cdots 1 n r 1 r 2 cdots r n nbsp Using distributive property P x an 1 x xn r1 r2 rn xn 1 1 n r1r2 rn rn 1 xn r1 r2 rn xn 1 1 n r1r2 rn displaystyle P x a n 1 x x n r 1 r 2 cdots r n x n 1 cdots 1 n r 1 r 2 cdots r n r n 1 x n r 1 r 2 cdots r n x n 1 cdots 1 n r 1 r 2 cdots r n nbsp After expanding and collecting like terms P x an 1xn 1 an 1 r1 r2 rn rn 1 xn 1 n 1 r1r2 rnrn 1 displaystyle begin aligned P x a n 1 x n 1 a n 1 r 1 r 2 cdots r n r n 1 x n cdots 1 n 1 r 1 r 2 cdots r n r n 1 end aligned nbsp The inductive hypothesis holds true for n 1 displaystyle n 1 nbsp therefore it must be true n N displaystyle forall n in mathbb N nbsp Conclusion anxn an 1xn 1 a1x a0 anxn an r1 r2 rn xn 1 1 n r1r2 rn displaystyle a n x n a n 1 x n 1 cdots a 1 x a 0 a n x n a n r 1 r 2 cdots r n x n 1 cdots 1 n r 1 r 2 cdots r n nbsp By dividing both sides both sides by an displaystyle a n nbsp it proves the Vieta s formulas true History editAs reflected in the name the formulas were discovered by the 16th century French mathematician Francois Viete for the case of positive roots In the opinion of the 18th century British mathematician Charles Hutton as quoted by Funkhouser 1 the general principle not restricted to positive real roots was first understood by the 17th century French mathematician Albert Girard 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 edit nbsp Mathematics portalContent algebra Descartes rule of signs Newton s identities Gauss Lucas theorem Properties of polynomial roots Rational root theorem Symmetric polynomial and elementary symmetric polynomialReferences edit Funkhouser 1930 Viete theorem Encyclopedia of Mathematics EMS Press 2001 1994 Funkhouser H Gray 1930 A short account of the history of symmetric functions of roots of equations American Mathematical Monthly 37 7 Mathematical Association of America 357 365 doi 10 2307 2299273 JSTOR 2299273 Vinberg E B 2003 A course in algebra American Mathematical Society Providence R I ISBN 0 8218 3413 4 Djukic Dusan et al 2006 The IMO compendium a collection of problems suggested for the International Mathematical Olympiads 1959 2004 Springer New York NY ISBN 0 387 24299 6 Retrieved from https en wikipedia org w index php title Vieta 27s formulas amp oldid 1209808406, wikipedia, wiki, book, books, library,