Let
be the sum of the products of distinct polynomial
roots
of the polynomial equation of degree

(1)

where the roots are taken at a time (i.e., is defined as the symmetric
polynomial )
is defined for , ..., . For example, the first few values of are
and so on. Then Vieta's formulas states that

(5)

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 seconddegree polynomial
by multiplying out,
so
Similarly, for a thirddegree polynomial,
so
See also
NewtonGirard Formulas,
Polynomial Discriminant,
Polynomial
Roots,
Symmetric Polynomial
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References
Bold, 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: SpringerVerlag, pp. 56, 1995.Coolidge,
J. L. A
Treatise on Algebraic Plane Curves. New York: Dover, pp. 12, 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: SpringerVerlag, 1993.Viète, F.
Opera
mathematica. 1579. Reprinted Leiden, Netherlands, 1646.Referenced
on WolframAlpha
Vieta's Formulas
Cite this as:
Weisstein, Eric W. "Vieta's Formulas."
From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/VietasFormulas.html
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