Concept
In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as Ruffini's rule), but the method can be generalized to division by 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, helping to prevent sign errors. The first example is synthetic division with only a monic linear denominator . The numerator can be written as . The zero of the denominator is . The coefficients of are arranged as follows, with the zero of on the left: The after the bar is "dropped" to the last row. The is multiplied by the before the bar, and placed in the . An is performed in the next column. The previous two steps are repeated and the following is obtained: Here, the last term (-123) is the remainder while the rest correspond to the coefficients of the quotient. The terms are written with increasing degree from right to left beginning with degree zero for the remainder and the result. Hence the quotient and remainder are: The above form of synthetic division is useful in the context of the polynomial remainder theorem for evaluating univariate polynomials. To summarize, the value of at is equal to the remainder of the division of by The advantage of calculating the value this way is that it requires just over half as many multiplication steps as naive evaluation. An alternative evaluation strategy is Horner's method. This method generalizes to division by any monic polynomial with only a slight modification with changes in bold. Using the same steps as before, perform the following division: We concern ourselves only with the coefficients. Write the coefficients of the polynomial to be divided at the top.
