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Probably Known Property of homogeneous Polynomials in R[X,Y]

Source: Moldova 2008 IMO-BMO First TST Problem 4

March 3, 2008
algebrapolynomialfactorialcalculusintegrationalgebra proposed

Problem Statement

A non-zero polynomial S∈R[X,Y] S\in\mathbb{R}[X,Y] is called homogeneous of degree d d if there is a positive integer d d so that S(\lambda x,\lambda y)\equal{}\lambda^dS(x,y) for any λ∈R \lambda\in\mathbb{R}. Let P,Q∈R[X,Y] P,Q\in\mathbb{R}[X,Y] so that Q Q is homogeneous and P P divides Q Q (that is, P∣Q P|Q). Prove that P P is homogeneous too.
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