= Rolle root count with multiplicities
{c}
{title2=$N(p')\ge N(p)-1$}
If a nonzero real <polynomial> has $N$ roots in an open real interval, counted with <multiplicity of a root>, its derivative has at least $N-1$ roots in that interval, also counted with multiplicity, provided $N\ge1$. Each root of multiplicity $m$ becomes a root of the derivative of multiplicity $m-1$. The <Rolle theorem> supplies one further root between each pair of distinct consecutive roots. Thus $k$ distinct roots of multiplicities $m_j$ give at least $\sum_j(m_j-1)+(k-1)=N-1$ derivative roots.
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