Vieta's Formulas

Vieta's formulas relate a polynomial's coefficients to symmetric sums of its polynomial roots. 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)