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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]], let $f \\in R[x]$ with $f \\neq 0$ and let $c \\in R$.\nThe multiplicity of $c$ as a root of $f$ is the largest integer $m \\ge 0$ such that $(x - c)^m$ divides $f$.\n\n## Remarks\n\nThe integer $m$ exists, because $(x - c)^m$ is monic of degree $m$, so it cannot divide $f$ for $m \u003e \\deg f$.\nThe multiplicity is positive exactly when $c$ is a [[root|00EHX6FWC0]] of $f$.\nA root of multiplicity $1$ is a simple root. A root of multiplicity at least $2$ is a repeated root.\nIn $\\mathbb{Q}[x]$, $f = (x - 1)^3 (x + 2)$ has the root $1$ with multiplicity $3$ and the root $-2$ with multiplicity $1$.\n\n## Explanations\n\n### L1 human\n\nThe multiplicity counts how many times the factor $x - c$ appears in $f$.\nFor $(x - 1)^2 (x + 3)$, the root $1$ counts twice and the root $-3$ counts once.\n",
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