Definition:Multiplicity (Polynomial)

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

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

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

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

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