Composition of polynomials

Abstract

The motivation to write this paper arose out of the following problem which was posed in a recent mathematical olympiad:Given a polynomial P(X) with integer coefficients, show that there exist non-zero polynomials Q(X), R(X) with integer coefficients such that P(X)Q(X) is a polynomial in X2 and P(X)R(X) is a polynomial in X3.For instance, if P(X) = 2 − 5X + 3X2 + 12X3, then we notice that Q (X) = 2 + 5X + 3X2 − 12X3 serves the purpose for the first part, viz.A moment's thought makes it fairly evident that this trick easily solves the first part of the problem for a general polynomial P(X).