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Putnam 1992 B4

Source: Putnam 1992

July 18, 2022
Putnamalgebrapolynomialderivative

Problem Statement

Let p(x)p(x) be a nonzero polynomial of degree less than 19921992 having no nonconstant factor in common with x3−xx^3 -x. Let d1992dx1992(p(x)x3−x)=f(x)g(x) \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 -x } \right) =\frac{f(x)}{g(x)} for polynomials f(x)f(x) and g(x).g(x). Find the smallest possible degree of f(x)f(x).
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