Problem

Source: Moldova TST 2023

Tags: algebra, polynomial



Polynomials $(P_n(X))_{n\in\mathbb{N}}$ are defined as: $$P_0(X)=0, \quad P_1(X)=X+2,$$$$P_n(X)=P_{n-1}(X)+3P_{n-1}(X)\cdot P_{n-2}(X)+P_{n-2}(X), \quad (\forall) n\geq2.$$Show that if $ k $ divides $m$ then $P_k(X)$ divides $P_m(X).$