Synthetic division
In algebra, synthetic division is a method of performing Euclidean division of polynomials, with less writing and fewer calculations than occur with polynomial long division. It is mostly taught for division by binomials of the form
but the method generalizes to division by any monic polynomial, and to any polynomial.
The advantages of synthetic division are that it allows one to calculate without writing variables, it uses few calculations, and it takes significantly less space on paper than long division. Also, the subtractions in long division are converted to additions by switching the signs at the very beginning, preventing sign errors.
Synthetic division for linear denominators is also called division through Ruffini's rule.
The first example is synthetic division with only a monic linear denominator .
Write the coefficients of the polynomial that is to be divided at the top (the zero is for the unseen 0x).
Negate the coefficients of the divisor.
Write in every coefficient of the divisor but the first one on the left.
Note the change of sign from −3 to 3. "Drop" the first coefficient after the bar to the last row.
Multiply the dropped number by the number before the bar, and place it in the next column.
Perform an addition in the next column.
Repeat the previous two steps and the following is obtained:
Count the terms to the left of the bar. Since there is only one, the remainder has degree zero. Mark the separation with a vertical bar.
The terms are written with increasing degree from right to left beginning with degree zero for both the remainder and the result.
The result of our division is:
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