# Synthetic Division

Pronunciation: /sɪnˈθɛt.ɪk dɪˈvɪ.ʒən/ Explain

Synthetic division is an algorithm for quickly dividing polynomials when the divisor is in the form x - r. Synthetic division is formally called Ruffini's rule. Synthetic division is commonly used to verify roots of a polynomial.

Example 1
StepEquation(s)Description
1 This is the dividend.
2 This is the divisor.
3 This is the operation that will be performed.
4 Identify the implied coefficients and fill them in. Applying the property of multiplying by 1 to the term x3 gives an implied coefficient is 1. Notice that there is no term x2. Applying the property of multiplying by 0 then the property of adding 0 to the x2 term gives 0 · x2.
5 Copy the coefficients into the top of a grid.
6 Copy the additive inverse of the constant term of the divisor\ into the grid on the left.
7 Copy down the coefficient of the term with the largest degree. In this case, the coefficient is 1.
8 Multiply the first number on the bottom by the number on the left. Place the result in the 2nd row, the next column to the right.
9 Add the numbers in the first two rows of the next column. Place the result in the 3rd row.
10 Multiply the number on the 3rd row of the current column by the number on the left. Place the result in the 2nd row, the next column to the right.
11 Add the numbers in the first two rows of the next column. Place the result in the 3rd row.
12 Multiply the number on the 3rd row of the current column by the number on the left. Place the result in the 2nd row, the next column to the right.
13 Add the numbers in the first two rows of the next column. Place the result in the 3rd row.
14 Multiply the number on the 3rd row of the current column by the number on the left. Place the result in the 2nd row, the next column to the right.
15 Copy the coefficients of the result into a polynomial.
16 Simplify the polynomial.
17 Here is the problem with the result.
18 Check the work by multiplying the\ result by the divisor.
Example 1
1. McAdams, David E.. All Math Words Dictionary, synthetic division. 2nd Classroom edition 20150108-4799968. pg 174. Life is a Story Problem LLC. January 8, 2015. Buy the book

McAdams, David E. Synthetic Division. 5/7/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/s/syntheticdivision.html.

### Revision History

5/7/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)