Division of a Polynomial by a Binomial / polynomial

Ruffini's Rule: The rule allows for the quick division of a polynomial f(x) by a binomial of the form (x-r) where where x is a variable and r is an integer. A polynomial of degree zero is called a constant, degree 1 a linear polynomial, degree 2 as a quadratic polynomial, degree 3 as a cubic polynomial, degree 4 as quartic polynomial etc. Zero polynomial is different from a polynomial of degree zero in the sense that its degree is either -1 or -infinity. Polynomial of one variable is called a univariate, of two variables a bivariate or more than 3 multivariate. A polynomial of more than 1 variable is called homogenous of degree n if all the terms are of degree n. A monic polynomial is a polynomial    in which the leading coefficient cn is equal to 1.

The division of a polynomial by a polynomial is through synthetic division.

Ruffini's rule is the special case of synthetic division where the divisor is a linear factor. The scheme was devised by Paolo Ruffini, an Italian mathematician of late Eighteenth and early Nineteenth century. Ruffini was born in Valentano,  a small town in Italy famous for its fortress and churches in the year 1765.At the age of 23, he earned University degree in philosophy, mathematics and medicine & surgery. He worked both as a doctor and a Professor of mathematics. He made significant contribution to the group theory, probability and computational mathematics. Ruffini died in 1822.

f (x) = x3 + x2 + x1 +x0

f1(x) = x -

f2(x) x2 + x1 +x0

f (x)/f1(x) = x2 + x1 + (quotient)

and                                                 (remainder)

f (x)/f2(x) = x1 + (quotient)

and                x1 + (remainder)