[1] The rule is a special case of synthetic division in which the divisor is a linear factor.
To divide P(x) by Q(x): 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 main problem is that Q(x) is not a binomial of the form x − r, but rather x + r. Q(x) must be rewritten as Now the algorithm is applied: So, if original number = divisor × quotient + remainder, then Ruffini's rule can be used when one needs the quotient of a polynomial P by a binomial of the form
the Euclidean division is written as This gives a (possibly partial) factorization of
The fundamental theorem of algebra states that every polynomial of positive degree has at least one complex root.
The above process shows the fundamental theorem of algebra implies that every polynomial p(x) = anxn + an−1xn−1 + ⋯ + a1x + a0 can be factored as where
The method was invented by Paolo Ruffini, who took part in a competition organized by the Italian Scientific Society (of Forty).
The challenge was to devise a method to find the roots of any polynomial.