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Date: 17-1-2019
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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
(2) |
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(3) |
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(4) |
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 second-degree polynomial by multiplying out,
(6) |
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(7) |
so
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Similarly, for a third-degree polynomial,
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so
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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: 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. Opera mathematica. 1579. Reprinted Leiden, Netherlands, 1646.
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