2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/134563In this paper we give an elementary proof of the Fundamental Theorem of Algebra for polynomials over the rational tropical semi-ring. We prove that, tropically, the rational numbers are algebraically closed. We provide a simple algorithm for factoring tropical polynomials of a single variable. A central idea is the concept of least-coefficient polynomials as representatives for classes of functionally equivalent polynomials. This idea has importance far beyond the proof of the Fundamental Theorem of Tropical Algebra.8 pagesCombinatorics20M14 (Primary); 52B20 (Secondary)text