##  [Vieta's Formulae](/vietas-formulae-0) 

 Definition

Algebraic identities linking the coefficients of a polynomial to the elementary symmetric functions of its roots; for a monic polynomial these give exact equalities between signed sums and products of the roots and the polynomial's coefficients.

 

 

 

 

 

 





## Principle

Principle

If a degree-n polynomial with leading coefficient a_n has roots r1,...,rn in an algebraic closure, then symmetric sums of the roots (sum of roots, sum of pairwise products, etc.) are expressible as ±(coefficient)/a_n with alternating signs according to degree.

 

 

 

 

 





## Demonstration

Demonstration

For ax^2+bx+c with roots r and s, Vieta's formulae state r+s=−b/a and rs=c/a. For x^3+px^2+qx+r with roots α,β,γ we have α+β+γ=−p, αβ+αγ+βγ=q, and αβγ=−r.

 

 

 

 

## Misapplication

Misapplication

Using Vieta's relations when the leading coefficient is zero (not a polynomial of that degree), ignoring multiplicities or working modulo a ring where division by the leading coefficient is invalid, or attempting to apply the relations to transcendental 'roots' outside an algebraic closure.

 

 

 

 

 





## Consequence

Consequence

Allows reconstruction of polynomial coefficients from symmetric functions of roots and provides a direct tool for expressing elementary symmetric invariants; it underlies elimination, constructive polynomial design, and many algebraic manipulations.

 

 

 

 

## Reversal

Reversal

Given a compatible set of elementary symmetric values one can form the monic polynomial having those values as its coefficients (up to sign), so Vieta's formulae invert to build polynomials from root-sum data; conversely, Newton's identities invert power sums to symmetric sums.

 

 

 

 

 





## Boundary

Boundary

Valid over any commutative ring where the operations make sense and where roots are taken in an ambient algebraic closure or extension; requires nonzero leading coefficient to divide coefficients if expressing relations with denominators; does not provide the explicit factorization unless roots exist in the working field.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Sits adjacent to Newton's identities (which relate power sums to symmetric sums) and to the theory of symmetric polynomials: Vieta's gives direct coefficient–root symmetric relations, while Newton's identities handle power-sum data and recursions.

 

 

 

 

 





## Synthesis

Synthesis

Vieta's formulae are the direct algebraic bridge between coefficients and roots: they express coefficients as signed elementary symmetric functions of the roots (and vice versa for monic polynomials), enabling polynomial reconstruction and algebraic relations among roots.