be the sum of the products of distinct polynomial
of the polynomial equation of degree
where the roots are taken at a time (i.e., is defined as the symmetric
is defined for , ..., . For example, the first few values of are
and so on. Then Vieta's formulas states that
The theorem was proved by Viète (also known as Vieta, 1579) for positive roots only, and the general theorem was proved by Girard.
This can be seen for a second-degree polynomial
by multiplying out,
Similarly, for a third-degree polynomial,
See alsoNewton-Girard Formulas
, Polynomial Discriminant
, Symmetric Polynomial
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ReferencesBold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 56,
1982.Borwein, P. and Erdélyi, T. "Newton's Identities."
§1.1.E.2 in Polynomials
and Polynomial Inequalities. New York: Springer-Verlag, pp. 5-6, 1995.Coolidge,
J. L. A
Treatise on Algebraic Plane Curves. New York: Dover, pp. 1-2, 1959.Girard,
A. Invention nouvelle en l'algèbre. Leiden, Netherlands: Bierens de
Haan, 1884.Hazewinkel, M. (Managing Ed.). Encyclopaedia
of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical
Encyclopaedia," Vol. 9. Dordrecht, Netherlands: Reidel, p. 416,
1988.van der Waerden, B. L. Algebra,
Vol. 1. New York: Springer-Verlag, 1993.Viète, F.
mathematica. 1579. Reprinted Leiden, Netherlands, 1646.
on Wolfram|AlphaVieta's Formulas
Cite this as:
Weisstein, Eric W. "Vieta's Formulas."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VietasFormulas.html