Synthetic Division (#2) - For All Polynomials

Originally posted by myself on a different blog, June 2015.

Synthetic Division for All Polynomials: Steps with Examples

Overview

• Rationale and Research
• General Directions
• Example 1: polynomial divided by x+B
• Example 2: polynomial divided by Ax+B
• Example 3: polynomial divided by x²+Bx+C
• Example 4: polynomial divided by Ax²+Bx+C
• Example 5: polynomial divided by a cubic polynomial

Research and Rationale

The synthetic division using a divisor beyond x minus some number is rarely taught. I took several books off my shelf by various authors to see what they taught about synthetic division.

When a polynomial is to be divided by a binomial of the form x-c, a shortcut process called synthetic division may be used. (Martin-Gay, 2013, pg 365)

To find the quotient as well as the remainder when a polynomial of degree 1 or higher is divided by x-c, a shortened version of long division, called synthetic division, makes the task simpler. (Sullivan, 2008, pg 57)

We can use synthetic division to divide polynomials if the divisor is of form x-c. This method provides a quotient more quickly than long division. (Blitzer, 2013, pg 448)

These texts represent how synthetic division is typically taught in the United States. It is taught that synthetic division can only be used with the divisor, x-c. In an article by Lainghou Fan, he wrote about textbooks:

However, the college algebra textbooks usually introduce the method to a polynomial … by a binomial of g(x)=x-c, without mentioning if this classical method can be applied when the divisor is a polynomial of degree being higher than 1, and some further explicitly stated that it is not applicable to such a divisor.

Then he quotes several textbooks by Larson, Hostetler, and Edwards (Fan, 2003, pg 1). For a while, I was only familiar with using synthetic division with a divisor of x-c. However, when one compares synthetic division with long division, the coefficients are the same (see figure 1). 

Since the numbers repeat, it would then make sense that it could be applied to any scenario of division of polynomials. When I asked the question about synthetic division for any divisor and searched out the topic, I found the article by Fan. In his article, he details how one would divide using synthetic division beyond x-c. In this post, I will attempt to show general directions with examples similar to the method that is typically found in textbooks.

General Directions

• Step 1: Set-up the board
• Top: arrange coefficients of dividend in descending order. If any missing coefficients, use zeroes.
• Side: arrange divisor top-down, make coefficients the opposite, and ignore the leading coefficient. If any missing coefficients, use zeroes.
• If there is a leading coefficient of the divisor that’s not one, include it on the side below the row. Place its reciprocal there.
• Put in the bars.  Put in the remainder box in the bottom-right corner. It’s the same number of columns as the rows on the side.
• Step 2: Drop down the first number
• Step 3: If there is a bottom side number, multiply it by the number in step 2
• Step 4: Form diagonal by multiplying the new number from step 3 with the side #’s. Put the results into the next column/s diagonally (i.e. “/”).
• Step 5: Add the column
• Step 6: If there is a bottom side number, multiply it by the result in step 5
• Step 7: Form diagonal by multiplying the new number from step 6 with the side #’s. Put the results into the next column/s diagonally (i.e. “/”).
• Step 8: Repeat steps 5-7 until the diagonal touches the right side. Once this happens, add the remaining columns; the result/s will go in the area for the remainder
• Step 9: Write the solution in descending order.
• The degree of the solution is the degree of dividend minus the degree of the divisor.
• The remainder in descending order over the divisor. The degree of the remainder is always one less degree than the divisor.

Note: step 3 and step 6 can be eliminated by removing the leading coefficient of divisor. Since it is a rational expression, you can divide each term of the dividend and the divisor by the leading coefficient of the divisor. 

Usually looking at steps can be difficult, so we will now look at some examples. With the examples, I will start with the less complex divisor to the more complex divisor. When we go through these examples, the step numbers will not exactly line up. The steps above are general directions. Some steps might be skipped (i.e. step 6), because they might not necessary. Also other steps might be repeated (i.e. steps 5-7 in step 8). The general directions for synthetic division will be made clear by the examples.

If you would like to see video examples of synthetic division, check out my Synthetic Division for All Polynomials playlist.

Examples

Example 1

Step 1: set-up

Step 2: bring down first number

Step 3: form diagonal by multiplying

Step 4: add columns

Step 5: form diagonal by multiplying

Step 6: add columns

Step 7: form diagonal by multiplying

Step 8: add columns

Example 2

Step 1: set-up

Step 2: bring down first number

Step 3: multiply by bottom side number

Step 4: form diagonal by multiplying

Step 5: add column

Step 6: multiply by bottom side number

Step 7: form diagonal by multiplying

Step 8: add column

Step 9: multiply by bottom side number

Step 10: form diagonal by multiplying








Step 11: add column

Example 3

Step 1: set-up








Step 2: bring down first number

Step 3: form diagonal by multiplying

Step 4: add column

Step 5: form diagonal by multiplying

Step 6: add column

Step 7: form diagonal by multiplying

Step 8: add columns

Example 4

Step 1: set-up

Step 2: bring first number

Step 3: multiply by bottom side number

Step 4: form diagonal by multiplying

Step 5: add column

Step 6: multiply by bottom side number

Step 7: form diagonal by multiplying

Step 8: add column

Step 9: multiply by bottom side number

Step 10: form diagonal by multiplying

Step 11: add columns

Example 5

Step 1: set-up

Step 2: bring down first number

Step 3: form diagonal by multiplying

Step 4: add column

Step 5: form diagonal by multiplying

Step 6: add column

Step 7: form diagonal by multiplying

Step 8: add columns

References

Blitzer, R. (2013). Algebra for college students (7th ed.). Boston, MA: Pearson.

Fan, Lianghou (2003). A generalization of synthetic division and a general theorem of division of polynomials. Mathematical Medley, 30(1), 30-37. Retrieved June 26, 2015, from http://eprints.soton.ac.uk/168861/1/FLH_article_on_polynomial_division.pdf

Martin-Gay, E. (2013). Intermediate algebra (Custom Ed. for Valencia College). Boston, MA: Pearson.

Sullivan, M. (2008). College Algebra (8th ed.). Upper Saddle River, NJ: Pearson Prentice Hall.