Vieta's Formulas

Vieta's formulas relate a polynomial's coefficients to symmetric sums of its polynomial roots. Let , ..., be the polynomial roots, counted with multiplicity, of the polynomial equation of polynomial degree

(1)

where . For , ..., , let be the elementary symmetric polynomial obtained by summing products of roots with distinct indices. The first few values are

			(2)
			(3)
			(4)

and so on. Then Vieta's formulas state that

(5)

Viète (1646) obtained these relations for positive roots, and Girard (1884) gave their general form.

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)

Conversely, prescribed values of , ..., determine the monic polynomial

(23)

Over , the system of equations obtained by fixing these elementary symmetric polynomials has as its solutions precisely all orderings of the multiset of roots of (Wu 2003). If the distinct roots have multiplicities , ..., , the number of distinct ordered solutions is the multinomial coefficient . In particular, there are distinct ordered solutions when every root is a simple root.