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P(x) = a_nx^n+a_{n-1}x^{n-1}+...+a_0 - All-Russian MO 2004 Regional (R4) 11.3

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September 27, 2024
algebrapolynomial

Problem Statement

Let the polynomial P(x)=anxn+an−1xn−1+...+a0P(x) = a_nx^n+a_{n-1}x^{n-1}+...+a_0 has at least one real root and a0≠0a_0 \ne 0. Prove that, consequently crossing out the monomials in the notation P(x)P(x) in some order, we can obtain the number a0a_0 from it so that each intermediate polynomial also has at least one real root.
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