In the previous two parts, we introduced the Restricted Synthetic Division (RSD) that is taught in classrooms and dealt with in textbooks and we extended it to a simple variation that is generally excluded from the domain of polynomial division subject to SD. In today’s post, we will introduce Universal SD (USD) for the general case of dividing any polynomial by any other polynomial of equal or lesser degree. Since it is far easier to use USD than it is to explain the procedure in the abstract in writing, I shall first present several specific examples with a description of what is being done, so when I describe the general procedure, the examples will clarify what is meant.

Example 1. Divide by .

(2 -3) 4 -5 + 7 +6 +8 -9 +8 -12 +2 -3 + 6 -9 -2 +3 4 +3 + 1 -1 (+3 -6)

The final array, given above, is produced by using the following algorithm. The repeated steps are indicated by using “prime” notation. To learn how to use this procedure, it’s best to construct the array for all three examples yourself by following the steps in the order given in the procedure.

1. Write the divisor, , as (2 -3) and all the coefficients of the dividend in the top row. Leave room for two (the degree of the divisor) intermediate rows, draw an addition bar and bring down the 4 (the first coefficient of the dividend) to the bottom (quotient) row.

2. Write +8 -12, the product of 4 and 2 -3, in the next two columns of the second row.

3. Write 3, the sum of the numbers in the second column, in the bottom row.

4. Repeat steps 2 and 3 to find products and sums in the rest of the array as follows:

2′. Write +6 -9, the product of +3 and 2 -3, in the next two columns of the third row.

3′. Write +1, the sum of the numbers in the third column, in the bottom row.

2”. Write +2 -3, the product of +1 and 2 -3, in the next two columns of the second row.

3”. Write -1, the sum of the numbers in the fourth column, in the bottom row.

2”’. Write -2 +3, the product of -1 and 2 -3, in the next two columns of the third row.

3”’. Write +3 -6, the sums of the numbers in the last two columns, in the bottom row. (Once there is an entry in the last column of an intermediate row, just write the sums in the remaining columns.)

5. Since the degree of the quotient is the degree of the dividend minus the degree of the divisor, the bottom row indicates that the quotient is () or with a remainder of .

Example 2. Divide by .

(-1 +4 -2) 2 +3 -4 +5 -12 -1 +13 -2 -2 +8 -4 -2 +8 -4 -1 +4 -2 +4 -16 +8 -3 +12 -6 2 +1 +3 +2 -4 (+5 -7 +6)

The final array, given above, is produced by using the following algorithm.

1. Write the divisor, , as (-1 +4 -2) and all the coefficients of the dividend in the top row. Leave room for three (the degree of the divisor) intermediate rows, draw an addition bar and bring down the 2 to the bottom (quotient) row.

2. Write -2 +8 -4, the product of 2 and -1 +4 -2, in the next three columns of the second row.

3. Write +1, the sum of the numbers in the second column, in the bottom row.

4. Repeat steps 2 and 3 to find products and sums in the rest of the array as follows:

2′. Write -1 +4 -2, the product of +1 and -1 +4 -2, in the next three columns of the third row.

3′. Write +3, the sum of the numbers in the third column, in the bottom row.

2”. Write -3 +12 -6, the product of +3 and -1 +4 -2, in the next three columns of the fourth row.

3”. Write +2, the sum of the numbers in the fourth column, in the bottom row.

2”’. Write -2 +8 -4, the product of +2 and -1 +4 -2, in the next three columns of the second row.

3”’. Write -4, the sum of the numbers in the fifth column, in the bottom row.

2””. Write +4 -16 +8, the product of -4 and -1 +4 -2, in the next three columns of the third row.

3””. Write +5 -7 +6, the sums of the numbers in the last three columns, in the bottom row. (Once there is an entry in the last column of an intermediate row, just write the sums in the remaining columns.)

5. Since the degree of the quotient is the degree of the dividend minus the degree of the divisor, the bottom row indicates that the quotient is or with a remainder of .

Example 3. Divide by .

(3 0 0 -2) 1 -5 +6 0 +2 -7 +9 +4 0 0 +5 3 0 0 -2 0 0 0 0 -6 0 0 +4 -9 0 0 +6 0 0 0 0 0 0 0 0 0 0 0 0 1 -2 0 0 0 -3 0 (+4 0 +6 +5)

The final array, given above, is produced by using the following algorithm. I won’t be as detailed as I was in the first two examples, but if you want to learn this procedure, I suggest that you supply the details yourself.

1. Write the divisor, as (3 0 0 -2) , writing zeros for the coefficients of the missing powers of *x,* and all the coefficients of the dividend in the top row, including zeros for the missing powers of *x*. Leave room for four (the degree of the divisor) intermediate rows, draw an addition bar and bring down the 1 to the bottom (quotient) row.

2. Write the product of that 1 and 3 0 0 -2 in the next four columns of the second row.

3. Write the sum of the numbers in the second column in the bottom row.

4. Repeat steps 2 and 3 to find and enter the products and sums in the rest of the array, using the same pattern of operations as in example 2. (Reminder: once an entry is made in the last column of an intermediate row, just write the sums in the remaining columns without doing any more multiplications.)

5. The bottom row indicates that the quotient is / or with a remainder of .

Here is the general procedure for the Universal Synthetic Division (USD) of any polynomial of degree *n*, of the form, , by any polynomial of degree *m*, of the form, , where *m* ≤ *n* and the leading coefficient is 1. (For leading coefficients , follow the instructions given in USD, Part 2 for dealing with that issue in RSD.)

* ….. ….. *

* ….. ….. * . ….. . …..

. …..

. …..

. …..

.

. ….. …..

(We cannot punctuate the bottom row to separate the coefficients of the quotient from the coefficients of the remainder without knowing what *m* and *n* equal. We also cannot put in the details of the right side of the intermediate rows, because without knowing what *m* and *n* equal, we don’t know which row will have an entry in the last column of the array.)

The final array, whose entries are labeled abstractly above, is produced by the following algorithm.

1. Write the negatives of the coefficients of the divisor (except for the leading coefficient), through , and all the coefficients of the dividend, through , in the top row, punctuated to distinguish the two sets of numbers. Any missing powers of the variable require writing zeros as the coefficients in the appropriate positions. Leaving room for *m* intermediate rows, insert an addition bar and bring down the number in the position to the position (leftmost position of the bottom (quotient) row).

2. Write the products of the most recent entry in the bottom row (the first time, the multiplier will be ) and through in the next *m* columns of the first intermediate row in which the column following that most recent entry is empty (the first time, it will be columns *n* – 1 through *n* – *m* in intermediate row 1).

3. Write the sum of the numbers in the column following the most recent bottom-row entry in the bottom (quotient) row of that column (the first time, it will be column *n* – 1).

4. Repeat steps 2 and 3. Stop this process as soon as you make an entry in the last column of an intermediate row, and write just the sums in the remaining columns.

5. The degree of the quotient is *n* – *m*, the coefficients of the quotient are the first *n* – *m* + 1 entries in the bottom row and the coefficients of the remainder are the last *m* entries of the bottom row. Any zeros in the bottom row indicate missing powers of the variable in corresponding positions in the quotient and/or the remainder.

COMMENTS

1. Every entry in the bottom row is the sum of and all the intermediate-row entries in that column (these are entries in the array that are labeled ). Every leftmost entry in intermediate row *j* is the product of and . Each intermediate row consists of one or more sets of *m* products of some *s* entry and through . More details about the notation can be teased out, but it doesn’t serve to clarify the procedure, so I will let those readers who are more compulsive than I am play with the notation and let me know when they are satisfied with their creations. I have already let my desire for a notation that is simultaneously simple, comprehensive and structurally sound delay my publishing this procedure for too long.

2. After making the first set of *m* entries in the last intermediate row, the entries of the intermediate rows form a matrix in the form of a rhombus with *m* entries on each side. If you get to make a complete second set of entries in the last intermediate row, you’ll have a matrix in the form of a parallelogram with 2*m* entries on the horizontal sides and *m* entries on the oblique sides.

3. RSD can now be seen as an instance of USD in which *m* = 1, and SD by can be seen as an instance of USD that has been abbreviated and compressed by omitting all unnecessary zeros.

4. While USD is more complicated than RSD in order to accommodate all the usual suspects in polynomial division, i.e., to make it *Universal*, it’s the only alternative to polynomial long division, and it’s much less writing, and in place of a four-step algorithm using the arithmetic operations of division, multiplication and subtraction, in USD we have a two-step algorithm using only multiplication and addition. After using it a few times, you’ll breathe a sigh of relief as the procedure becomes automatic.