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The factorization of \(f(x)x^{ n }+g(x)\) with \(f(x)\) monic and of degree \(\leq 2\). (English. French summary) Zbl 1293.11049

Let \(f(X)=X^2+bX+c\in Z[X]\) with \(c\geq2\), \(|b|<2\sqrt{c-1}\), and let \(g\in Z[X]\), \(\deg g=t\). The author shows that under certain assumptions on the coefficients of \(g\) (depending on \(f\)) the polynomial \(f(X)X^n+g(X)\) is reducible for all \(n > n_0(f,t)\). This leads to a series of corollaries asserting irreducibility for certain families of polynomials.

MSC:

11C08 Polynomials in number theory
12E05 Polynomials in general fields (irreducibility, etc.)
26C10 Real polynomials: location of zeros
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References:

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[2] A. Brauer, On the irreducibility of polynomials with large third coefficient II. Amer. J. Math. 73 (1951), 717-720. · Zbl 0042.25201
[3] J.B. Conway, Functions of One Complex Variable. New York: Springer-Verlag. · Zbl 0887.30003
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