In mathematics, Ruffini's rule is a method for computation of the Euclidean division of a polynomial by a binomial of the form x − r. It was described by Paolo Ruffini in 1809. The rule is a special case of synthetic division in which the divisor is a linear monic factor.
Algorithm The rule establishes a method for dividing the polynomial:
P ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 {\displaystyle P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}
by the binomial:
Q ( x ) = x − r {\displaystyle Q(x)=x-r}
to obtain the quotient polynomial:
R ( x ) = b n − 1 x n − 1 + b n − 2 x n − 2 + ⋯ + b 1 x + b 0 . {\displaystyle R(x)=b_{n-1}x^{n-1}+b_{n-2}x^{n-2}+\cdots +b_{1}x+b_{0}.}
The algorithm is in fact the long division of P(x) by Q(x). To divide P(x) by Q(x):
Take all the coefficients of P(x), including zero for any missing terms, and write them down in order of their decreasing degrees. Then, write r at the bottom-left edge just over the line:
a n a n − 1 … a 1 a 0 r {\displaystyle {\begin{array}{c|c c c c|c}&a_{n}&a_{n-1}&\dots &a_{1}&a_{0}\\r&&&&&\\\hline &&&&&\\\end{array}}}
Pass the leftmost coefficient (an) to the bottom just under the line.
a n a n − 1 … a 1 a 0 r a n = b n − 1 {\displaystyle {\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}}}
Multiply the rightmost number under the line by r, and write it over the line and one position to the right.
… excerpt ends here. Continue reading the full article.
