Definition:Horner's Rule

Definition

Let $\map p x$ be a polynomial.

$\map p x = a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0$

Then $\map p x$ can be expressed in the following form:

$\map p x = \paren {\cdots \paren {\paren {a_n x + a_{n - 1} } x + a_{n - 2} } x + \cdots + a_1} x + a_0$

Also known as

Some sources refer to Horner's rule as Horner's method, but that name is also given to the Ruffini-Horner method.

This technique is also known as nested multiplication.

Examples

Arbitrary Example

The polynomial:

$\map p x = 4 x^3 - 2 x^2 + 3 x - 1$

can be expressed using Horner's rule as:

$\map p x = \paren {\paren {4 x - 2} x + 3} x - 1$

Also see

• Results about Horner's rule can be found here.

Source of Name

This entry was named for William George Horner.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Horner's rule
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Horner's method: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Horner's method: 1.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Horner's rule
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Horner's rule (Horner's method)