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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]], let $f = a_n x^n + \\dots + a_1 x + a_0 \\in R[x]$ with $n \\ge 0$, and let $c \\in R$.\nHorner's method computes the elements $b_n, b_{n-1}, \\dots, b_0$ of $R$ by\n$$b_n = a_n, \\qquad b_k = a_k + c \\, b_{k+1} \\quad (k = n - 1, \\dots, 0),$$\nand returns $b_0$.\n\n## Remarks\n\nThe output is the value: $b_0 = f(c)$, see [[polynomial evaluation|0GJ7T0NK49]].\nMoreover $f = (x - c)(b_n x^{n-1} + \\dots + b_2 x + b_1) + b_0$, so the method also gives the quotient of $f$ by $x - c$.\nExample: for $f = 2x^3 - 6x^2 + 2x - 1$ and $c = 3$, one gets $b_3 = 2$, $b_2 = 0$, $b_1 = 2$, $b_0 = 5$.\nCheck: $f(3) = 54 - 54 + 6 - 1 = 5$, and $f = (x - 3)(2x^2 + 2) + 5$.\nNon-example: computing each power $c^k$ separately and then the sum gives the same value, but it is not Horner's method and uses more multiplications.\nThe method uses $n$ multiplications and $n$ additions.\n\n## Explanations\n\n### L1 human\n\nHorner's method reads the coefficients from the top and, at each step, multiplies by $c$ and adds the next coefficient.\n",
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