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Sequence of polynomials eventually has real roots

Source: IMC 2007 Day 2 Problem 6

August 7, 2007
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Problem Statement

Let f≠0 f \ne 0 be a polynomial with real coefficients. Define the sequence f0,f1,f2,… f_{0}, f_{1}, f_{2}, \ldots of polynomials by f0=f f_{0}= f and fn+1=fn+fn′ f_{n+1}= f_{n}+f_{n}' for every n≥0 n \ge 0. Prove that there exists a number N N such that for every n≥N n \ge N, all roots of fn f_{n} are real.
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