Viète's formulas

In mathematics, Vieta's formulas are formulas that relate the coefficients of a polynomial to sums and products of its roots. Named after François Viète (more commonly referred to by the Latinised form of his name, Franciscus Vieta), the formulas are used specifically in algebra.

Any general polynomial of degree n

(with the coefficients being real or complex numbers and a_n ≠ 0) is known by the fundamental theorem of algebra to have n (not necessarily distinct) complex roots x₁, x₂, ..., x_n. Vieta's formulas relate the polynomial's coefficients { a_k } to signed sums and products of its roots { x_i } as follows:

Equivalently stated, the (n − k)th coefficient a_n−k is related to a signed sum of all possible subproducts of roots, taken k-at-a-time:

for k = 1, 2, ..., n (where we wrote the indices i_k in increasing order to ensure each subproduct of roots is used exactly once).

The left hand sides of Vieta's formulas are the elementary symmetric functions of the roots.

Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R. In this case the quotients ${\displaystyle a_{i}/a_{n}}$ belong to the ring of fractions of R (or in R itself if ${\displaystyle a_{n}}$ is invertible in R) and the roots ${\displaystyle x_{i}}$ are taken in an algebraically closed extension. Typically, R is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers.

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