[b]Ruffini's rule[/b] is a shortcut method for dividing a polynomial by a linear factor of the form [math]x-a[/math], which can be used in place of the standard long division algorithm. It is sometimes taken as synthetic division, but this is just a particular case.[br][list][*]This method reduces the polynomial and the linear factor into a set of numeric values.[/*][*]After these values are processed, the resulting set of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.[/*][/list]

[list][*]Write both, dividend and divisor, in descending order according to variable’s degree.[/*][*]If a power of x is missing from the dividend, a term with that power and a zero coefficient must be inserted into the correct position in the polynomial.[/*][*]All the variables and their exponents are removed from the dividend, leaving only a list of the dividend's coefficients.[/*][*]Because only the constant term [math]a[/math] of the linear factor [math]x-a[/math] is necessary for Ruffini's rule, the divisor is modified into a one-term sequence [math]a[/math] (if the divisor was [math]x+a[/math], rewriting as [math]x-(-a)[/math] would result in a modified divisor sequence of [math]-a[/math] instead).[/*][*]The first number in the dividend is put into the first position of the result area (below the horizontal line). This number is the coefficient of the [math]x^n[/math] term in the original dividend polynomial.[/*][*]Then this first entry in the result is multiplied by the divisor [math]a[/math] and the product is placed under the next term in the dividend.[/*][*]The number from the dividend and the result of the multiplication are added together and the sum is put in the next position on the result line.[/*][*]This process is continued for the remainder of the numbers in the dividend.[/*][*]All numbers except the last become the coefficients of the quotient polynomial. The last entry in the result list is the remainder.[/*][/list]

[url=https://mathworld.wolfram.com/topics/Stover.html]Stover, Christopher[/url]. "Ruffini's Rule." From [url=https://mathworld.wolfram.com/][i]MathWorld[/i][/url]--A Wolfram Web Resource, created by [url=https://mathworld.wolfram.com/about/author.html]Eric W. Weisstein[/url]. [url=https://mathworld.wolfram.com/RuffinisRule.html]https://mathworld.wolfram.com/RuffinisRule.html[/url]