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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $c \\in R$.\nThe evaluation homomorphism at $c$ is the map\n$$\\operatorname{ev}_c \\colon R[x] \\to R, \\qquad \\operatorname{ev}_c(f) = f(c),$$\nwhere $f(c)$ is the [[value of $f$ at $c$|0GJ7T0NK49]].\nIt is a [[ring homomorphism|0GGE0Z0JQ1]].\n\n## Remarks\n\nThe map $\\operatorname{ev}_c$ is surjective, because $\\operatorname{ev}_c(r) = r$ for each constant $r \\in R$.\nIts [[kernel|0AAHE7GEW6]] is the principal ideal $(x - c)$. This is a form of the [[factor theorem|0K3MWA7DMC]].\nThe commutativity of $R$ is necessary: in general $(f g)(c) \\neq f(c) g(c)$ when $c$ does not commute with the coefficients.\n\n## Explanations\n\n### L1 human\n\nEvaluation at $c$ turns each polynomial into an element of $R$.\nIt respects sums and products: to evaluate $f g$, you can evaluate $f$ and $g$ and multiply.\n",
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