{
  "aliases": [
    "polynomial-remainder-theorem"
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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]], let $f \\in R[x]$ and let $c \\in R$.\nThen the remainder of the [[division|0MES3KS0ZC]] of $f$ by $x - c$ is the constant polynomial $f(c)$.\nThat is, there is $q \\in R[x]$ with $f = (x - c) q + f(c)$.\n\n## Remarks\n\nThe proof: division by the monic polynomial $x - c$ gives $f = (x - c) q + r$ with $r$ a constant. [[Evaluation|0GJ7T0NK49]] at $c$ gives $r = f(c)$.\nFor example, the remainder of $x^3 - 2x + 5$ divided by $x - 2$ is $2^3 - 4 + 5 = 9$.\n\n## Explanations\n\n### L1 human\n\nYou do not need long division to find the remainder of $f$ divided by $x - c$.\nYou only compute $f(c)$.\n",
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    "aliases": [
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    "name": "Remainder theorem",
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  "markdown": "https://math2.tamnd.com/node/0TEDQH56RW.md",
  "msc": [
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  "name": "Remainder theorem",
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