# Definition:Horner's Method

## Disambiguation

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**Horner's Method** may refer to:

### Ruffini-Horner Method

The **Ruffini-Horner method** is a technique for finding the roots of polynomial equations in $1$ real variable.

Let $E_0$ be the polynomial equation in $x$:

- $\map p x = 0$

Suppose that a root $x_0$ being sought is the positive real number expressed as the decimal expansion:

- $x_0 = \sqbrk {abc.def}$

The process begins by finding $a$ by inspection.

We then form a new equation $E_1$ whose roots are $100 a$ less than those of $E_0$.

This will have a root $x_1$ in the form:

- $x_1 = \sqbrk {bc.def}$

Similarly, $b$ is found by inspection.

We then form a new equation $E_2$ whose roots are $10 b$ less than those of $E_1$.

The process continues for as many digits accuracy as required.

### Horner's Rule

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:

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

## Source of Name

This entry was named for William George Horner.