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
May 03, 2024
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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 a n x n a n 1 x n 1 a 1 x a 0 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 r 1 r 2 r n 1 r n a n 1 a n r 1 r 2 r 1 r 3 r 1 r n r 2 r 3 r 2 r 4 r 2 r n r n 1 r n a n 2 a n r 1 r 2 r n 1 n a 0 a n 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 i 1 lt i 2 lt lt i k n j 1 k r i j 1 k a n k a n 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 a i a n displaystyle a i a n nbsp belong to the field of fractions of R and possibly are in R itself if a n displaystyle a n nbsp happens to be invertible in R and the roots r i 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 a n displaystyle a n nbsp is not a zero divisor and P x displaystyle P x nbsp factors as a n x r 1 x r 2 x r n 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 x 2 1 displaystyle P x x 2 1 nbsp has four roots 1 3 5 and 7 Vieta s formulas are not true if say r 1 1 displaystyle r 1 1 nbsp and r 2 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 r 1 1 displaystyle r 1 1 nbsp and r 2 7 displaystyle r 2 7 nbsp or r 1 3 displaystyle r 1 3 nbsp and r 2 5 displaystyle r 2 5 nbsp Example editVieta s formulas applied to quadratic and cubic polynomials The roots r 1 r 2 displaystyle r 1 r 2 nbsp of the quadratic polynomial P x a x 2 b x c displaystyle P x ax 2 bx c nbsp satisfyr 1 r 2 b a r 1 r 2 c a 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 r 1 r 2 r 3 displaystyle r 1 r 2 r 3 nbsp of the cubic polynomial P x a x 3 b x 2 c x d displaystyle P x ax 3 bx 2 cx d nbsp satisfyr 1 r 2 r 3 b a r 1 r 2 r 1 r 3 r 2 r 3 c a r 1 r 2 r 3 d a 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 equalitya n x n a n 1 x n 1 a 1 x a 0 a n x r 1 x r 2 x r n 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 r 1 r 2 r n 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 r 1 x r 2 x r n displaystyle x r 1 x r 2 cdots x r n nbsp the terms are precisely 1 n k r 1 b 1 r n b n x k displaystyle 1 n k r 1 b 1 cdots r n b n x k nbsp where b i displaystyle b i nbsp is either 0 or 1 accordingly as whether r i displaystyle r i nbsp is included in the product or not and k is the number of r i displaystyle r i nbsp that are included so the total number of factors in the product is n counting x k displaystyle x k nbsp with multiplicity k as there are n binary choices include r i displaystyle r i nbsp or x there are 2 n 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 r i displaystyle r i nbsp for xk all distinct k fold products of r i displaystyle r i nbsp As an example consider the quadraticf 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 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 a 2 a 2 displaystyle a 2 a 2 nbsp a 1 a 2 r 1 r 2 displaystyle a 1 a 2 r 1 r 2 nbsp and a 0 a 2 r 1 r 2 displaystyle a 0 a 2 r 1 r 2 nbsp with which we can for example identify r 1 r 2 a 1 a 2 displaystyle r 1 r 2 a 1 a 2 nbsp and r 1 r 2 a 0 a 2 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 r 1 r 2 r n displaystyle r 1 r 2 dots r n nbsp and complex coefficients a 0 a 1 a n displaystyle a 0 a 1 dots a n nbsp where a n 0 displaystyle a n neq 0 nbsp Then the inductive hypothesis is thatP x a n x n a n 1 x n 1 a 1 x a 0 a n x n a n r 1 r 2 r n x n 1 1 n a n r 1 r 2 r n 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 a 2 a 1 displaystyle a 2 a 1 nbsp be coefficients of the quadratic and a 0 displaystyle a 0 nbsp be the constant term Similarly let r 1 r 2 displaystyle r 1 r 2 nbsp be the roots of the quadratic a 2 x 2 a 1 x a 0 a 2 x r 1 x r 2 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 a 2 x 2 a 1 x a 0 a 2 x 2 r 1 x r 2 x r 1 r 2 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 a 2 x 2 a 1 x a 0 a 2 x 2 r 1 r 2 x r 1 r 2 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 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 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 a n 1 x n 1 a n x n a 1 x a 0 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 r n 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 r 1 r 2 r n displaystyle r 1 r 2 cdots r n nbsp P x x r n 1 a n 1 x n 1 a n x n a 1 x a 0 x r n 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 a n 1 displaystyle a n 1 nbsp the leading coefficient P x displaystyle P x nbsp from the polynomial in the square brackets P x a n 1 x r n 1 x n 1 a n x n a n 1 a 1 a n 1 x a 0 a n 1 x r n 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 a n 1 x r n 1 x n z n 1 x n 1 z 0 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 a n 1 x r n 1 x n r 1 r 2 r n x n 1 1 n r 1 r 2 r n 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 a n 1 x x n r 1 r 2 r n x n 1 1 n r 1 r 2 r n r n 1 x n r 1 r 2 r n x n 1 1 n r 1 r 2 r n 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 a n 1 x n 1 a n 1 r 1 r 2 r n r n 1 x n 1 n 1 r 1 r 2 r n r n 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 a n x n a n 1 x n 1 a 1 x a 0 a n x n a n r 1 r 2 r n x n 1 1 n r 1 r 2 r n 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 a n 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 portal Content 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,