What is Synthetic Division? (With Examples)

student studying math

I am here to help you understand another algebra tool that you might already know. That tool is synthetic division. What you might not understand is the power this tool holds.

Synthetic division is a shortcut method of dividing a binomial into a polynomial. Using the coefficients of the polynomial, we go through a process of multiplying and adding until we reach the final coefficient. One result of synthetic division is that it helps you determine if the binomial is a polynomial factor. Synthetic division is also be used in conjunction with the remainder theorem to evaluate that polynomial at a value x=c.

This is a tool that can help make your math life easier as you get into more complex math classes. Master it now so that it can pay off later. Let’s take a look at how it works.

Breaking it down.

There are two methods of dividing a binomial into a polynomial. The first method that you may be familiar with is long division. Long division looks similar to the long division we learned in grade school. The process is similar, except that you divide a binomial into a polynomial. Here is an example of what long division looks like.

The other method of division is synthetic division. Synthetic division is the shorter version of long division. Most students appreciate how it condenses the previous method into shorter steps while still getting the same results.

Before we dive into performing synthetic division, we will look at its basic structure.

We start with the divisor on the left-hand side, as we can see above. We use a bracket to close it off from the next section. The divisor will come from the binomial that we divide into our polynomial. So if our polynomial is f(x)=x3-4x2+3x+5 and we are dividing the binomial, (x-3), into it, our divisor would be 3.

The line of coefficients that appears next to the divisor is from our polynomial. One important thing about this line of coefficients is that you must list them in order from highest to lowest power. Once they are listed in order, you can take each coefficient and place it in the synthetic division setup.

The last thing that we have to consider before performing any steps is what to do with missing powers in our polynomial. If your polynomial is missing any power, you must replace that value with a zero in your listing. If you have the polynomial, f(x)= x4+3×2-5x+3, you will list the missing x3 value as a zero. The graphic below shows you an example.

Now we are ready to continue to perform the steps of synthetic division.

Working through synthetic division, step by step.

Now we can get into the nitty-gritty steps of performing synthetic division. Once we have everything set up, we can begin the calculations.

Here is a condensed version of the synthetic division steps.

  • Carry down the first coefficient and place it below the line.
  • Multiply by the divisor and place the result under the coefficient in the next column.
  • Add the two values in that column and place the result underneath the line.
  • Take the new value and multiply it by the divisor again and place the result in the next column.

We will keep multiplying and adding until we have reached the last column where the last coefficient is located. I know that sounds like a lot, but let’s break it down even further.

The first step is to bring the first coefficient down below the line.

We take that coefficient that we dropped down, and we multiply it by our divisor. Once we get the result, we can place it in the second column in the space above the line and right below the 8.

We then take the two numbers, 8 and 1, and add them. The result goes under the line in the second column.

We then repeat the process. We take the 9 and multiply it with the 1 again. We then place the result in the third column, above the line, and right below the 17.

We then add the numbers 17 and 9 and place the result in the third column below the line. We then repeat the process of multiplying and adding until we reach the last column.

The result in that last column contains a lot of important information. If your goal is to find out if your divisor is a root factor or a zero of your polynomial, then hopefully, you got a zero in that last column. If no zero appears, then that value is not a zero. In our case, 1 is a root factor or a zero of this polynomial.

If your goal is to use the remainder theorem with synthetic division, then the last column gives valuable information too. The last value in the final column shows our polynomial evaluated at f(c). In our case, plugging 1 into f(x) will evaluate to 0, or f(1)=0.

I have included a very short video of all the slides explaining the process.

Example problems.

In the next two very short videos, I use synthetic division to find the roots or zeros of complex polynomials.

I included these to show the process of taking the results after performing synthetic division and continuing the process until we can’t factor the polynomial anymore. I think this shows how robust this process is and how amazing synthetic division is.

As always, keep practicing. You only get better if you keep doing the exercises consistently. ๐Ÿ˜


Just a nerd who loves math. Trying to help the world one problem at a time.

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