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Definition: Multiplicity of a Root of a Polynomial
Let $p\in R[X]$ be a polynomial over a ring. Replacing the variable $x$ by some element of the ring makes the polynomial a function $p:R\to R.$ If $b\in R$ is a zero of this function, i.e. $p(b)=0,$ then there exists a natural number $n\in\mathbb N,$ $n\ge 1$ such that:
 $(xb)^n\mid p$ (the polynomial $(xb)^n$ is a divisor of $p$) and
 $(xb)^{n+1}\not\mid p$ (the polynomial $(xb)^{n+1}$ is not a divisor of $p$).
Such a natural number is called the multiplicity of the root $b$ of the polynomial $p.$
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References
Bibliography
 Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013