2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75251In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coefficients, the general quintonomial coefficient as a sum of the binomial and quadrinomial coefficients, and gives a general determination of the coefficients of the expansion of any polynomial (1+x+x^2+...+x^m)^n as a sum of the coefficients of lower degree polynomial coefficients.8 pages. Seems to be first English translation of Euler's Latin original ``De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc.)^n, Nova Acta Academiae Scientarum Imperialis Petropolitinae 12 (1801), 47-57History and OverviewCombinatorics01A50; 05A10text