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
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\$.
%0 Generic
%1 citeulike:3036252
%A Euler, Leonhard
%D 2007
%K Vor1750 available-in-tex-format field-theory mathematics pre1750
%T 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
%U http://arxiv.org/abs/0707.0699
%X 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\$.
@misc{citeulike:3036252,
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\$.},
added-at = {2009-08-02T17:14:35.000+0200},
archiveprefix = {arXiv},
author = {Euler, Leonhard},
biburl = {https://www.bibsonomy.org/bibtex/2f1c600c83d141eb56e0e867732e9c95b/rwst},
citeulike-article-id = {3036252},
citeulike-linkout-0 = {http://arxiv.org/abs/0707.0699},
citeulike-linkout-1 = {http://arxiv.org/pdf/0707.0699},
description = {my bookmarks from citeulike},
eprint = {0707.0699},
interhash = {7da9e9a370f2023774b1fc3eaba449c4},
intrahash = {f1c600c83d141eb56e0e867732e9c95b},
keywords = {Vor1750 available-in-tex-format field-theory mathematics pre1750},
month = Jul,
posted-at = {2008-07-23 08:37:53},
priority = {2},
timestamp = {2009-08-06T10:17:07.000+0200},
title = {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},
url = {http://arxiv.org/abs/0707.0699},
year = 2007
}