Synthetic Division

Synthetic division is a shortcut method for dividing one polynomial by another which can be used in place of the standard long division algorithm. This method reduces the dividend and divisor polynomials into a list of numeric values. After these values are processed, the resulting list of numeric outputs is used to construct the polynomial quotient and the polynomial remainder.

For an example of synthetic division, consider dividing by . First, if a power of is missing from either polynomial, a term with that power and a zero coefficient must be inserted into the correct position in the respective polynomial. In this case, the term is missing from the dividend while the term is missing from the divisor; therefore, is added between the quintic and the cubic terms of the dividend while is added between the cubic and the linear terms of the divisor:

and

respectively.

Next, all the variables and their exponents () are removed from the dividend, leaving instead a list consisting solely of its coefficients: , , , , , and . This sequence of numbers is placed into a division-like configuration.

Variables are then similarly removed from the divisor, yielding a sequence , , , and . Because the divisor fails to be monic, one must keep track of the leading coefficient ( in this case); after doing so, the divisor's leading coefficient is discarded and the signs of its remaining coefficients are "reversed," thereby leaving a "modified sequence" of , , and corresponding to the divisor. This modified sequence and the leading coefficient are placed in the division-like configuration as illustrated above.

The first number in the dividend ( in this case) is put into the first position of the first result area (i.e., the first row below the horizontal line). This number is the coefficient of the term in the original dividend polynomial.

At this point, it is imperative that the leading coefficient of the divisor be recognized; before continuing, the first number of the dividend () must be divided by this leading coefficient (), the result of which () will be recorded into the first position of the second result area (i.e., the second row below the horizontal line). This number is the modified coefficient of the term in the original dividend after division by the leading coefficient of the divisor.

Now, the first entry in this latest result () is multiplied by each element of the coefficient sequence from the divisor (, , and ), the products of which are placed diagonally under the next dividend terms as illustrated above.

As the algorithm progresses, the numbers from the dividend are added systematically to the results of the multiplications performed; in particular, this addition happens when the resulting product element is directly above the horizontal line. The result of the addition is placed on the first result line.

At this point, the process repeats. The most recently entered number on the first result line ( in this case) is divided by the leading coefficient of the divisor () to yield a number ( here) placed on the second result line.

This result () is multiplied by the left-hand divisor sequence (, , and ) to yield products (, , and ) which are placed diagonally under the subsequent dividend terms.

Next comes addition along the subsequent column (the dividend column, here consisting of , , and ), the result of which ( in this case) is divided by the leading divisor coefficient () to yield a result of .

Finally, the process repeats: is multiplied by the sequence , , to yield the sequence , , , which again is placed diagonally beneath the corresponding dividend terms.

Adding the subsequent column (the dividend column consisting of , , , ) yields a result (namely, ), and because any subsequent collections of products would consist of more numbers (namely, numbers, one for each of the three left-hand sequence numbers) than there are dividend terms remaining (there are two such terms, namely the -coefficient of 2 and the constant, ), no additional products are required. Hence, this result () need not be divided by the leading divisor coefficient (), whereby the remaining columns may be summed without division. This final step is illustrated above.

The result is the list of six numbers (the leftmost three from the second result row and the rightmost three from the first result row), namely

The number of coefficients of the polynomial quotient is , where and are the degrees of the dividend and divisor, respectively, and . Here, a degree-5 polynomial is divided by a degree-3 polynomial, so the first three numbers of the displayed result sequence, namely 2, 0, and , are the coefficients of the polynomial quotient . The remaining positions give the coefficients of the polynomial remainder , whose degree is less than . Hence, the polynomial quotient has the form:

Moreover, the remaining numbers (namely, , , and ) of (3) correspond to the coefficients of the polynomial remainder ; here,

The polynomial quotient and polynomial remainder can be combined into a single expression:

Note, too, that the only division operations performed during this computation consisted of dividing entries from the first result row by the leading divisor coefficient , whereby it follows that the polynomial quotient was computed after performing only three divisions.

The result can be verified directly:

In particular, multiplying the polynomial quotient by the divisor and adding the polynomial remainder yields the original dividend polynomial, thus confirming the validity of the result.

The process described above is a general form of synthetic division for univariate polynomials. It is variously called generalized synthetic division, expanded synthetic division, or generalized expanded synthetic division. Terminology varies. The case in which the divisor is a monic polynomial of arbitrary polynomial degree (no greater than the polynomial degree of the dividend) is sometimes called expanded synthetic division. The unqualified term "synthetic division" most often means the case of a monic linear divisor, which is more formally known as Ruffini's rule.