Ruffini's rule

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 1804. The rule is a special case of synthetic division in which the divisor is a linear factor. The rule establishes a method for dividing the polynomial: by the binomial: to obtain the quotient polynomial: The algorithm is in fact the 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: Pass the leftmost coefficient (an) to the bottom just under the line. Multiply the rightmost number under the line by r, and write it over the line and one position to the right. Add the two values just placed in the same column. Repeat steps 3 and 4 until no numbers remain. 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. Here is an example of polynomial division as described above. Let: P(x) will be divided by Q(x) using Ruffini's rule. 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: Write down the coefficients and r. Note that, as P(x) didn't contain a coefficient for x, 0 is written: | 2 3 0 | -4 | |