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real parts of roots of P(x) < n-1/2, (x-n+1) does not divide P(x), P(n) is prime

Source: Moldova 2002 TST 3 P4

August 25, 2018

complex numberspolynomialIrreducibleDivisibilityalgebra

Problem Statement

Let

P(x)

be a polynomial with integer coefficients for which there exists a positive integer n such that the real parts of all roots of

P(x)

are less than

n- \frac{1}{2}

, polynomial

x-n+1

does not divide

P(x)

, and

P(n)

is a prime number. Prove that the polynomial

P(x)

is irreducible (over

Z[x]