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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:

• $(x-b)^n\mid p$ (the polynomial $(x-b)^n$ is a divisor of $p$) and
• $(x-b)^{n+1}\not\mid p$ (the polynomial $(x-b)^{n+1}$ 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

1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013