Ruffini's rule is a shortcut method for dividing a polynomial by a linear polynomial of the form 
which can be used in place of the standard long
division algorithm. This method reduces the polynomial and the linear polynomial
into a list of numeric values. After these values are processed,
the resulting list of numeric outputs is used to construct
the polynomial quotient and the polynomial
remainder.
Note that Ruffini's rule is a special case of the more generalized notion of synthetic division in which the divisor polynomial is a monic linear polynomial. Confusingly, Ruffini's rule is sometimes referred to as synthetic division, thus leading to the common misconception that the scope of synthetic division is significantly smaller than that of the long division algorithm.
For an example of Ruffini's rule, consider 
divided by 
. First, if a power of 
is missing from the dividend,
a term with that power and a zero coefficient must be inserted into the correct position
in the polynomial. In this case the 
term is missing from the dividend,
so 
must be added between the cubic
and linear terms:
 |
(1)
|
Next, all the variables and their exponents (
, 
, 
)
are removed from the dividend, leaving only a list
of the dividend's coefficients:
3, 0, -6, and 2. Next, because only the polynomial
root (
)
of the linear polynomial 
is necessary for Ruffini's rule, the divisor is modified
into a one-term "sequence" 
. Note that if the divisor were 
, rewriting as 
would result in a modified divisor sequence
of 
instead.
The numbers representing the divisor and the dividend sequences are placed into a division-like configuration.
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 
term in the original dividend polynomial.
Then this first entry in the result (
) is multiplied by the divisor (
) and the product is placed under
the next term in the dividend (
).
Next 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.
The result is the sequence 3, 6, 6, 14. All numbers except the last become the coefficients of the polynomial quotient. Since a cubic polynomial was divided by a linear polynomial, the polynomial quotient is a 2nd-degree polynomial:
 |
(2)
|
The last entry in the result list (namely, 
) is the polynomial remainder.
The polynomial quotient and polynomial
remainder can be combined into one expression:
 |
(3)
|
(Note that no division operations were performed to compute the answer to this division problem.)
To verify that this process has worked, one can multiply the polynomial quotient by the divisor and add the polynomial remainder to obtain the original dividend polynomial:
 |  |  |
(4)
|
 |  |  |
(5)
|
Ruffini's rule can be used in conjunction with the polynomial remainder theorem to evaluate a polynomial at a real number. For example, consider the polynomial
 |
(6)
|
To find the value of 
,
the polynomial remainder theorem states
that 
is the polynomial remainder when 
is divided by 
.
Using Ruffini's rule gives the result illustrated above, so 
.
