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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]], let $f \\in R[x]$ with $f \\neq 0$ and let $c \\in R$.\nThen $c$ is a root of $f$ with [[multiplicity|03XVP8Q4GY]] at least $2$ if and only if $f(c) = 0$ and $f'(c) = 0$.\nHere $f'$ is the [[formal derivative|0AD4VS02H3]] of $f$.\n\n## Remarks\n\nThe proof: write $f = (x - c) g$. The product rule gives $f' = g + (x - c) g'$, so $f'(c) = g(c)$. And $g(c) = 0$ exactly when $(x - c)^2$ divides $f$.\nFor example, $f = x^3 - 3x + 2 = (x - 1)^2 (x + 2)$. Here $f(1) = 0$ and $f'(1) = 3 - 3 = 0$.\n\n## Explanations\n\n### L1 human\n\nAt a repeated root, the graph of a real polynomial touches the axis and has a horizontal tangent.\nSo the polynomial and its derivative are both zero there.\nThe algebraic statement is true over every commutative ring.\n",
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