Ruffini's rule
In mathematics, Ruffini's rule is an efficient technique for dividing a polynomial by a binomial of the form x − r. It was described by Paolo Ruffini in 1804. Ruffini's rule is a special case of synthetic division when 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):
1. Take the coefficients of P(x) and write them down in order. Then write r at the bottom left edge, just over the line:
2. Pass the leftmost coefficient (an) to the bottom, just under the line:
3. Multiply the rightmost number under the line by r and write it over the line and one position to the right:
4. Add the two values just placed in the same column
5. 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. As shown in the polynomial remainder theorem, this remainder is equal to P(r), the value of the polynomial at r.
Ruffini's rule has many practical applications; most of them rely on simple division (as demonstrated below) or the common extensions given still further below.
A worked example of polynomial division, as described above.
Let:
We want to divide P(x) 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. We must rewrite Q(x) in this way:
Now we apply the algorithm:
1. Write down the coefficients and r. Note that, as P(x) didn't contain a coefficient for x, we've written 0:
2. Pass the first coefficient down:
3. Multiply the last obtained value by r:
4. Add the values:
5. Repeat steps 3 and 4 until we've finished:
So, if original number = divisor × quotient + remainder, then
...
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