Horner's Method

A method for finding roots of a polynomial equation $f(x)=0$ by finding an equation whose roots are the same, but diminished by $r$, so

$$0=f(x+r)=f(r)+xf'(r)+{\textstyle{1\over 2}}x^2f''(r)+{\textstyle{1\over 3}}x^3f'''(r)+\ldots.$$

The expressions for $f(r)$, $f'(r)$, ... are then found by writing the coefficients $A$, $B$, ..., $F$ in a horizontal row, and letting a new letter shown as a denominator stand for the sum immediately above it. To find a root, first determine the integer part of the root through whatever means are needed, then reduce the equation by this amount. This gives the second digit, by which the equation is once again reduced (after suitable multiplication by 10) to find the third digit, and so on. Horner's process really boils down to the construction of a Divided Difference table.

See also Newton's Method