Synthetic Division: Definition, Steps & Examples

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  • 0:00 Synthetic Division
  • 0:48 Steps
  • 3:39 Another Example
  • 5:07 Finding Roots
  • 5:37 Lesson Summary
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Lesson Transcript
Instructor: Shaun Ault

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

This lesson is about synthetic division. Synthetic division helps us work out polynomial division when the divisor is simple. It also is a great shortcut to finding roots of polynomials.

Synthetic Division

We all love shortcuts! Synthetic division is a shortcut method for dividing a polynomial by a simple divisor of the form (x - n). The divisor must be of that form in order for synthetic division to work. If it's not, you'll have to use long division.

So why would we want to divide polynomials anyway? If a polynomial p(x) can be divided by (x - n) with no remainder, then x = n must be a zero or root of p(x). That is, a solution to the equation p(x) = 0. We'll see how that works in some examples below.

Synthetic division uses only the coefficients of a polynomial, or the constants in front of each x-term, so it saves a ton of writing compared to using long division of polynomials.

Steps

There is a specific order of steps to synthetic division; once you get the pattern, you will be off and running! The best way to explain these steps is by way of an example. Let's use synthetic division to work out this problem:

(3x^3 - x - 7) / (x - 2)

Step 1. If the divisor is (x - n), write n, and then draw a vertical line to the right of it. In this example, n = 2, so we would write:

2 |

Step 2. Put the coefficients of each x-term to the right of the vertical line. Start with the leading coefficient, that is, the coefficient of the highest power term, then place the coefficients of each lower degree term in descending order. It's very important to place a number for each degree, so if a particular term x^d does not show up, you should place a 0 in that spot. In our example, we could interpret 3x^3 - x - 7 = 3x^3 + 0x^2 + (-1)x + (-7). So the first line of your synthetic division work should look like this:

2 | 3 0 -1 -7

Step 3. Draw a horizontal line under the coefficients, leaving one empty row below the coefficients for work.


Synthetic Division step 1


Step 4. Bring the first coefficient down below the horizontal line, which is the easiest step of all!


Synthetic Division step 2


Step 5. Multiply the divisor number n by the number below the horizontal line, and place the result above the line below the next coefficient. In our running example. 2 * 3 = 6, which goes right below the 0.


Synthetic Division step 3


Step 6. Add the column to get the next coefficient in your answer. In our example, that would be 0 + 6 = 6. This new number goes below the horizontal line.


Synthetic Division step 4


Step 7. Repeat steps 5 and 6, filling each column from left to right until you get to the end of the coefficients. Do you know how the rest of the steps work out?


Synthetic Division steps


Circle or otherwise mark in some way the very last sum in the far right column. That number is the remainder , or the number left over after dividing.


Synthetic Division final


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