A double demonstration of a theorem of Newton, which gives a relation between the coefficient of an algebraic equation and the sums of the powers of its roots

TL;DR

This paper presents two proofs of Newton's identities relating polynomial coefficients to sums of roots' powers, one via logarithmic differentiation and another through induction, highlighting fundamental algebraic relationships.

Contribution

It provides two novel proofs of Newton's identities, enhancing understanding of algebraic relations between coefficients and roots.

Findings

Two proofs of Newton's identities are given.

The first proof uses logarithmic differentiation.

The second proof employs induction and root-product relations.

Abstract

Translation from the Latin original, "Demonstratio gemina theorematis Neutoniani, quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem" (1747). E153 in the Enestrom index. In this paper Euler gives two proofs of Newton's identities, which express the sums of powers of the roots of a polynomial in terms of its coefficients. The first proof takes the derivative of a logarithm. The second proof uses induction and the fact that in a polynomial of degree $n$, the coefficient of $x^{n-k}$ is equal to the sum of the products of $k$ roots, times $(-1)^k$.