Vieta's formulas

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)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}

(with the coefficients being real or complex numbers and a_n ≠ 0) is known by the fundamental theorem of algebra to have n (not necessarily distinct) complex roots x₁, x₂, ..., x_n. Vieta's formulas relate the polynomial's coefficients { a_k } to signed sums and products of its roots { x_i } as follows:

{\begin{cases}x_{1}+x_{2}+\dots +x_{n-1}+x_{n}=-{\dfrac {a_{n-1}}{a_{n}}}\\(x_{1}x_{2}+x_{1}x_{3}+\cdots +x_{1}x_{n})+(x_{2}x_{3}+x_{2}x_{4}+\cdots +x_{2}x_{n})+\cdots +x_{n-1}x_{n}={\dfrac {a_{n-2}}{a_{n}}}\\{}\quad \vdots \\x_{1}x_{2}\dots x_{n}=(-1)^{n}{\dfrac {a_{0}}{a_{n}}}.\end{cases}}

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

\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}}=(-1)^{k}{\frac {a_{n-k}}{a_{n}}}

for k = 1, 2, ..., n (where we wrote the indices i_k 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 $a_{i}/a_{n}$ belong to the ring of fractions of R (or in R itself if $a_{n}$ is invertible in R) and the roots $x_{i}$ 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 are only valid when $a_{n}$ is a non-zerodivisor and $P(x)$ factors as $a_{n}(x-x_{1})(x-x_{2})\dots (x-x_{n})$ . For example, in the ring of the integers modulo 8, the polynomial $P(x)=x^{2}-1$ has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, $x_{1}=1$ and $x_{2}=3$ , because $P(x)\neq (x-1)(x-3)$ . However, $P(x)$ does factor as $(x-1)(x-7)$ and as $(x-3)(x-5)$ , and Vieta's formulas hold if we set either $x_{1}=1$ and $x_{2}=7$ or $x_{1}=3$ and $x_{2}=5$ .

Example

Vieta's formulas applied to quadratic and cubic polynomial:

For the second degree polynomial (quadratic) $P(x)=ax^{2}+bx+c$ , roots $x_{1},x_{2}$ of the equation $P(x)=0$ satisfy

x_{1}+x_{2}=-{\frac {b}{a}},\quad x_{1}x_{2}={\frac {c}{a}}.

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

For the cubic polynomial $P(x)=ax^{3}+bx^{2}+cx+d$ , roots $x_{1},x_{2},x_{3}$ of the equation $P(x)=0$ satisfy

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}}.

Proof

Vieta's formulas can be proved by expanding the equality

a_{n}x^{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})

(which is true since $x_{1},x_{2},\dots ,x_{n}$ 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 $(x-x_{1})(x-x_{2})\cdots (x-x_{n}),$ the terms are precisely $(-1)^{n-k}x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}x^{k},$ where $b_{i}$ is either 0 or 1, accordingly as whether $x_{i}$ is included in the product or not, and k is the number of $x_{i}$ that are excluded, so the total number of factors in the product is n (counting $x^{k}$ with multiplicity k) – as there are n binary choices (include $x_{i}$ or x), there are $2^{n}$ terms – geometrically, these can be understood as the vertices of a hypercube. Grouping these terms by degree yields the elementary symmetric polynomials in $x_{i}$ – for x^k, all distinct k-fold products of $x_{i}.$

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 (Funkhouser), 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.

References

Hazewinkel, Michiel, ed. (2001) [1994], "Viète theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Funkhouser, H. Gray (1930), "A short account of the history of symmetric functions of roots of equations", American Mathematical Monthly, Mathematical Association of America, 37 (7): 357–365, JSTOR 2299273, doi:10.2307/2299273
Vinberg, E. B. (2003), A course in algebra, American Mathematical Society, Providence, R.I, ISBN 0-8218-3413-4

Djukić, Dušan; 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

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