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Part of 2011 India National Olympiad

Problems(1)

Polynomials P(x), Q(x) with a_n-b_n as prime [INMO 2011]

Source:

2/6/2011
Let P(x)=anxn+an−1xn−1+⋯+a0P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0 and Q(x)=bnxn+bn−1xn−1+⋯+b0Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0 be two polynomials with integral coefficients such that an−bna_n-b_n is a prime and anb0−a0bn≠0,a_nb_0-a_0b_n\neq 0, and an−1=bn−1.a_{n-1}=b_{n-1}. Suppose that there exists a rational number rr such that P(r)=Q(r)=0.P(r)=Q(r)=0. Prove that r∈Z.r\in\mathbb Z.
algebrapolynomialcalculusintegrationRational Root Theoremnumber theory