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Non-real roots of polynomial X^{n+1}-X^2+aX+1

Source: AIMO 6/1, German TST 2010

July 16, 2011
algebrapolynomialalgebra unsolved

Problem Statement

Let a∈R.a \in \mathbb{R}. Show that for n≥2n \geq 2 every non-real root zz of polynomial Xn+1−X2+aX+1X^{n+1}-X^2+aX+1 satisfies the condition ∣z∣>1nn.|z| > \frac{1}{\sqrt[n]{n}}.
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