Vieta's formulas

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For a method for computing π, see Viète's formula.

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[edit]

Basic formulas[edit]

Any general polynomial of degree n

P(x)=a_nx^n  + a_{n-1}x^{n-1} +\cdots + a_1 x+ a_0 \,

(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:

\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}

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:

\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}

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[edit]

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[edit]

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[edit]

Vieta's formulas can be proved by expanding the equality

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)

(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 xk, all distinct k-fold products of x_i.

History[edit]

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.

See also[edit]

References[edit]

  • 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