An irreducibility criterion for the sum of two relatively prime polynomials

We partly extend a result of Cavachi and Bonciocat on the sum of two relatively prime polynomials and prove that a polynomial of the form f (X) + Ng(X), where f(X) , g(X) is an element of Z[X] are two non-zero relatively prime polynomials with deg f < 1/2 deg g , is irreducible over Q for all but finitely many square-free positive integers N . 1 In addition, we derive a necessary and sufficient condition for a polynomial r + p(2)g (X) is an element of Z [X] to be reducible over Q for a sufficiently large prime number p .