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

			(2)
			(3)
			(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)
			(7)

so

			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

Similarly, for a third-degree polynomial,

			(14)
			(15)

so

			(16)
			(17)
			(18)
			(19)
			(20)
			(21)
			(22)

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.