Vieta's formulas

In mathematics, more specifically in algebra, Viète's Laws, named after François Viète, are formulas which relate the coefficients of a polynomial to signed sums and products of its roots.

The Laws

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.

Viète'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}={\tfrac {-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}={\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 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).

Generalization to rings

Viète's formulas hold more generally for polynomials with coefficients in any integral domain, as long as the leading coefficient a_n is invertible (so that the divisions make sense) and the polynomial has n distinct roots in that ring. The condition of being an integral domain is needed to assure that a polynomial of degree n cannot have more than n roots, and that if it has n roots then it is determined (up to a scalar) by those roots. However, if one replaces the assumption that x₁, …, x_n are the roots of the polynomial by the simpler requirement that one has the relation

{\begin{aligned}&{}\quad 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}),\end{aligned}}

then Viète's formulas even hold in any commutative ring, and they merely express the way the coefficients on the left hand side are formed when expanding the product on the right. Note that the given relation has to be required even in the case of an integral domain, if one wishes to allow some of the roots to coincide, and therefore needs to specify what multiplicity should be associated to each root.

Example

Viète'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

Viète's formulas can be proven 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 through the factors on the right-hand side, and identifying the coefficients of each power of $X.$

Formally, if one expands out $(X-x_{1})(X-x_{2})\cdots (X-x_{n}),$ the terms are precisely $(-1)^{n-k}x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}X^{k},$ where $a_{i}$ 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 terms 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 of X 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

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

Vinberg, E. B. (2003), A course in algebra, American Mathematical Society, Providence, R.I, ISBN 0821834134

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 0387242996 {{citation}}: Cite has empty unknown parameter: |coauthors= (help); Explicit use of et al. in: |first= (help)