1992 Putnam B4

Let $p(x)$ be a nonzero polynomial of degree less than 1992

having no nonconstant factor in common with $x^3 - x$. Let

$$

\frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) =

\frac{f(x)}{g(x)}

$$

for polynomials $f(x)$ and $g(x)$. Find the smallest possible degree of $f(x)$.

设 $p(x)$ 为次数小于 1992 的非零多项式

与 $x^3 - x$ 没有共同的非常数因子。让

$$

\frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 - x} \right) =

\frac{f(x)}{g(x)}

$$

对于多项式$f(x)$ 和$g(x)$。求 $f(x)$ 的最小可能次数。