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P_n(X)=P_{n-1}(X)+3P_{n-1}(X)\cdot P_{n-2}(X)+P_{n-2}(X)

Source: Moldova TST 2023

April 7, 2023
algebrapolynomial

Problem Statement

Polynomials (Pn(X))n∈N(P_n(X))_{n\in\mathbb{N}} are defined as: P_0(X)=0,   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),   (\forall) n\geq2. Show that if k k divides mm then Pk(X)P_k(X) divides Pm(X).P_m(X).
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