Horner's Method of Synthetic Division

Polynomial division with remainder is a building block for many important algebraic algorithms. Horner's method of synthetic division provides an efficient means of computing such quotients and remainders.

Given polynomials $f (x)$ and $g (x)$ in indeterminate $x$ , we will compute polynomials $q (x)$ (the quotient) and $r (x)$ (the remainder) such that $f (x) = g (x) q (x) + r (x)$ where either $r (x) = 0$ or the degree of $r (x)$ is smaller than the degree of $g (x)$ .

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Specifically, the Horner's method table is created as follows:

List the coefficients of $f (x)$ along the top row. These are listed from the leading coefficient down to the trailing coefficient.
Place $g (x)$ 's leading coefficient to the left of $f (x)$ 's leading coefficient. [The first row is now completed.]
Negate the remaining coefficients of $g (x)$ and list them (vertically) below $g (x)$ 's leading coefficient. [The first column is now completed.]
Draw a vertical line to the right of the $g (x)$ 's data. Draw a horizontal line well below this data. [Note: It is possible to have more rows than the number of coefficients of $g (x)$ . Also, it is possible to have less (non-empty) rows than than the number of $g (x)$ 's coefficients.]
In the next (uncompleted) column, sum up the entries in that column and divide by $g (x)$ 's leading coefficient. Place this number below the horizontal line at the bottom of that column. [This column is now completed.]
Take the number we just computed and multiply it by each of the negated coefficients of $g (x)$ (this is all of the coefficients of $g (x)$ except for the leading coefficient). Place these values in the next available (uncompleted) row. Starting in the next available uncompleted column. [Now this row is completed.]
This process should continue until our multiples fill a row to its end.
The remaining entries (below the horizontal line) at the bottom of uncompleted columns are computed by summing the entries in the column above. [This finishes the table.]

The bottom of the table just below the horizontal line is a list of coefficients of the quotient and remainder. The first

(\deg (f (x)) - \deg (g (x)) + 1)

numbers are the coefficients of the quotient

q (x)

(listed from leading to trailing coefficient) and the remaining numbers are the coefficients of the remainder

r (x)

(also listed from top to bottom).

Note: The polynomials can have coefficients in Sage's "symbolic ring" (i.e. SR). So coefficients such as "1+sqrt(2)" and "5+3*i" are allowed. The Sage code tries to clean up inputs (Ex: It changes (x-1)(x-2) to (x-1)*(x-2) and 3x to 3*x), but it cannot anticipate all syntax errors. As warning, check how your input was interpreted. It may also mis-correct certain inputs.

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Wikipedia entry for Horner's method.