Vieta,s Formulas
المؤلف:
Bold, B
المصدر:
Famous Problems of Geometry and How to Solve Them. New York: Dover
الجزء والصفحة:
...
23-2-2019
965
Vieta's Formulas
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 second-degree polynomial by multiplying out,
so
Similarly, for a third-degree polynomial,
so
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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