Multiplicity of a root

Definition

Let $R$ be a commutative ring, let $f \in R[x]$ with $f \neq 0$ and let $c \in R$. The multiplicity of $c$ as a root of $f$ is the largest integer $m \ge 0$ such that $(x - c)^m$ divides $f$.

Remarks

The integer $m$ exists, because $(x - c)^m$ is monic of degree $m$, so it cannot divide $f$ for $m > \deg f$. The multiplicity is positive exactly when $c$ is a root of $f$. A root of multiplicity $1$ is a simple root. A root of multiplicity at least $2$ is a repeated root. In $\mathbb{Q}[x]$, $f = (x - 1)^3 (x + 2)$ has the root $1$ with multiplicity $3$ and the root $-2$ with multiplicity $1$.

Explanations

L1 human

The multiplicity counts how many times the factor $x - c$ appears in $f$. For $(x - 1)^2 (x + 3)$, the root $1$ counts twice and the root $-3$ counts once.