Definition:Multiplicity (Polynomial)

Definition
Let $R$ be a commutative ring with unity.

Let $f \in R[X]$ be a nonzero polynomial.

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

The multiplicity of $a$ in $P$ is the largest positive integer such that $(x-a)^n$ divides $f(x)$ in $R[x]$.

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