Horner's method

Builds on Commutative ring Polynomial ring Polynomial evaluation

Definition

Let $R$ be a commutative ring, let $f = a_n x^n + \dots + a_1 x + a_0 \in R[x]$ with $n \ge 0$ , and let $c \in R$ . Horner's method computes the elements $b_n, b_{n-1}, \dots, b_0$ of $R$ by $b_n = a_n, \qquad b_k = a_k + c \, b_{k+1} \quad (k = n - 1, \dots, 0),$ and returns $b_0$ .

Remarks

The output is the value: $b_0 = f(c)$ , see polynomial evaluation. Moreover $f = (x - c)(b_n x^{n-1} + \dots + b_2 x + b_1) + b_0$ , so the method also gives the quotient of $f$ by $x - c$ . Example: for $f = 2x^3 - 6x^2 + 2x - 1$ and $c = 3$ , one gets $b_3 = 2$ , $b_2 = 0$ , $b_1 = 2$ , $b_0 = 5$ . Check: $f(3) = 54 - 54 + 6 - 1 = 5$ , and $f = (x - 3)(2x^2 + 2) + 5$ . Non-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. The method uses $n$ multiplications and $n$ additions.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Group concept
Ring definition
Commutative ring definition
Polynomial ring definition
Polynomial evaluation definition
Horner's method definition

Tag	`Z2EWA10YMB`
URI	`m2://Z2EWA10YMB`
Kind	definition
Writing	W0 stub
MSC	68W30, 13F20
Defines	Horner's method
Uses	Commutative ring, Polynomial ring, Polynomial evaluation
Changed	2 November 2026

Horner's method

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here