Editing Ruffini's rule (section)

==Algorithm==
The rule establishes a method for dividing the polynomial:
:<math>P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0</math>
by the binomial:
:<math>Q(x)=x-r</math>
to obtain the quotient polynomial:
:<math>R(x)=b_{n-1}x^{n-1}+b_{n-2}x^{n-2}+\cdots+b_1x+b_0.</math>

The algorithm is in fact the [[polynomial long division|long division]] of ''P''(''x'') by ''Q''(''x'').

To divide ''P''(''x'') by ''Q''(''x''):

# Take the coefficients of ''P''(''x'') and write them down in order. Then, write ''r'' at the bottom-left edge just over the line:
#:<math>
\begin{array}{c|c c c c|c}
 & a_n & a_{n-1} & \dots & a_1 & a_0\\
r & & & & & \\
\hline
 & & & & & \\
\end{array}
</math>
# Pass the leftmost coefficient (''a''<sub>''n''</sub>) to the bottom just under the line.
#:<math>
\begin{array}{c|c c c c|c}
 & a_n & a_{n-1} & \dots & a_1 & a_0\\
r & & & & & \\
\hline
 & a_n & & & & \\
 & =b_{n-1} & & & &
\end{array}
</math>
# Multiply the rightmost number under the line by ''r'', and write it over the line and one position to the right.
#:<math>
\begin{array}{c|c c c c|c}
 & a_n & a_{n-1} & \dots & a_1 & a_0\\
r & & b_{n-1} \cdot r & & & \\
\hline
 & a_n & & & & \\
 & =b_{n-1} & & & &
\end{array}
</math>
# Add the two values just placed in the same column.
#:<math>
\begin{array}{c|c c c c|c}
 & a_n & a_{n-1} & \dots & a_1 & a_0\\
r & & b_{n-1}\cdot r & & & \\
\hline
 & a_n & b_{n-1}\cdot r+a_{n-1} & & & \\
 & =b_{n-1} & =b_{n-2} & & &
\end{array}
</math>
# Repeat steps 3 and 4 until no numbers remain.
#:<math>
\begin{array}{c|c c c c|c}
 & a_n & a_{n-1} & \dots & a_1 & a_0 \\
r & & b_{n-1}\cdot r & \dots & b_1\cdot r & b_0 \cdot r \\
\hline
 & a_n & b_{n-1} \cdot r+a_{n-1} & \dots & b_1 \cdot r+a_1 & a_0+b_0 \cdot r \\
 & =b_{n-1} & =b_{n-2} & \dots & =b_0 & =s \\
\end{array}
</math>

The ''b'' values are the coefficients of the result (''R''(''x'')) polynomial, the degree of which is one less than that of ''P''(''x''). The final value obtained, ''s'', is the remainder. The [[polynomial remainder theorem]] asserts that the remainder is equal to ''P''(''r''), the value of the polynomial at ''r''.