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Monic polynomial with two high zero coefficients

Source: Polish Second Round 2001

March 6, 2012
algebrapolynomialalgebra proposed

Problem Statement

Let n≥3n\ge 3 be a positive integer. Prove that a polynomial of the form xn+an−3xn−3+an−4xn−4+…+a1x+a0,x^n+a_{n-3}x^{n-3}+a_{n-4}x^{n-4}+\ldots +a_1x+a_0, where at least one of the real coefficients a0,a1,…,an−3a_0,a_1,\ldots ,a_{n-3} is nonzero, cannot have all real roots.
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