Definition:Multiplicity (Polynomial)

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Definition

Let $R$ be a commutative ring with unity.

Let $P \in R \left[{X}\right]$ be a nonzero polynomial.

Let $a \in R$ be a root of $P$.


The multiplicity of $a$ in $P$ is the largest positive integer $n$ such that $\left({X - a}\right)^n$ divides $P \left({X}\right)$ in $R \left[{X}\right]$.



A double root is a root of multiplicity at least $2$.