1999 Putnam B2

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P''(x)$,

where $Q(x)$ is a quadratic polynomial and $P''(x)$ is the second

derivative of $P(x)$. Show that if $P(x)$ has at least two distinct

roots then it must have $n$ distinct roots.

令 $P(x)$ 为 $n$ 次多项式，使得 $P(x)=Q(x)P''(x)$，

其中 $Q(x)$ 是二次多项式，$P''(x)$ 是第二个

$P(x)$ 的导数。证明如果 $P(x)$ 至少有两个不同的

那么它必须有 $n$ 个不同的根。