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Polynomial - odd or even coefficients

Source: Romanian TST 2 2007, Problem 1

April 15, 2007

algebrapolynomialalgebra proposed

Problem Statement

Let

f = X^{n}+a_{n-1}X^{n-1}+\ldots+a_{1}X+a_{0}

be an integer polynomial of degree

n \geq 3

such that

a_{k}+a_{n-k}

is even for all

k \in \overline{1,n-1}

and

a_{0}

is even. Suppose that

f = gh

, where

g,h

are integer polynomials and

\deg g \leq \deg h

and all the coefficients of

h

are odd. Prove that

f

has an integer root.

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