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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]], let $f = a_0 + a_1 x + \\dots + a_n x^n \\in R[x]$ and let $c \\in R$.\nThe value of $f$ at $c$ is the element\n$$f(c) = a_0 + a_1 c + a_2 c^2 + \\dots + a_n c^n \\in R.$$\n\n## Remarks\n\nThe value $f(c)$ does not depend on the choice of $n$, because the extra coefficients are $0$.\nFor $f = x^2 - 2$ in $\\mathbb{Z}[x]$, $f(3) = 7$.\nIf $R$ is commutative, then $(f g)(c) = f(c) g(c)$ and $(f + g)(c) = f(c) + g(c)$. This gives the [[evaluation homomorphism|0125QVNCFQ]].\nHorner's rule computes $f(c) = a_0 + c(a_1 + c(a_2 + \\dots + c a_n))$ with $n$ multiplications.\n\n## Explanations\n\n### L1 human\n\nTo evaluate $f$ at $c$, you put $c$ in place of $x$ and compute.\nFor $f = x^2 + 1$ and $c = 2$, you get $f(2) = 5$.\n",
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