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x^{n+2}+ax^{n+1}+bx^{n}+a+b cannot be decomposed

Source: Korean Mathematical Olympiad 1995, Final Round P5 FKMO

July 21, 2018
polynomialInteger Polynomialgreatest common divisorprimefactorizationalgebra

Problem Statement

Let a,ba,b be integers and pp be a prime number such that: (i) pp is the greatest common divisor of aa and bb; (ii) p2p^2 divides aa. Prove that the polynomial xn+2+axn+1+bxn+a+bx^{n+2}+ax^{n+1}+bx^{n}+a+b cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than 11.
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