Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Prove it Ireland

Let $p(x)$ be a polynomial with rational coefficients. Prove that there exists a positive integer $n$ such that the polynomial $q(x)$ defined by
$q(x) = p(x + n) - p(x)$
has integer coefficients.

Solution

Each term in $p(x)$ is of the form $a_i x^i$ , where $a_i$ is rational. Expanding the expression $a_i(x+n)^i - a_i x^i$ , we see that $n$ is a factor in all terms. Thus it suffices to pick $n$ to equal the least common multiple of the denominators of the coefficients $a_i$ .

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