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given a polynomial of degree 4

Source: Vietnam TST 1994 for the 35th IMO, problem 3

June 25, 2005
algebrapolynomialalgebra unsolved

Problem Statement

Let P(x)P(x) be given a polynomial of degree 4, having 4 positive roots. Prove that the equation (1−4⋅x)⋅P(x)x2+(x2+4⋅x−1)⋅P′(x)x2−P′′(x)=0(1-4 \cdot x) \cdot \frac{P(x)}{x^2} + (x^2 + 4 \cdot x - 1) \cdot \frac{P'(x)}{x^2} - P''(x) = 0 has also 4 positive roots.
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