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Synthetic Division & Long Division of Polynomials

Synthetic Division & Long Division of Polynomials

Author: Sophia Tutorial

This lesson covers synthetic division and long division of polynomials.

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What's Covered

  • Polynomial Long Division
  • Synthetic Division 

Synthetic Division & Long Division of Polynomials

Polynomial Long Division

Polynomial long division sounds like a complicated process, and it can be without an understanding of the similarities between the long division from your elementary school days.  The process, or algorithm - the steps we take to arrive at our solution - is the same, we are just dealing with more complicated terms. 

Here is an example of polynomial long division.  As you read through the steps, think about how the process is similar to numeric long division: 


At this point, we repeat our steps of dividing, multiplying, subtracting, and bringing down the next term:

We repeat the cycle of steps once again:


This tells us that the quotient is 2 x squared minus 3 x plus 4 with zero remainder.


Synthetic Division

Synthetic division is a process that is designed to be easier than polynomial long division.  It works primarily with the coefficients to the term, and simplifies the steps in the algorithm to arrive at the same solution.  We are going to do the same division example as above, but set the problem up using synthetic division:

First, use the coefficients of the dividend (what is being divided) and write them inside of a box, as shown below:


It is important that these coefficients represent terms that are written in descending order of their degree.  If a polynomial is "missing a term" (for example, x3 + x has no x2 term), we must use 0 as a place holder when writing the coefficients in the box above.  This will ensure that the coefficients we get in our answer match to the proper term in the quotient. 

Next, we write in the value for "a" in our general (x – a) form for the divisor (what we are dividing by).  


Because we generally say (x – a), if the factor we are dividing by has a plus sign, the value of "a" is actually negative.

Now we have our synthetic division all set up.  In order to solve using synthetic division, we follow these steps:

  • Bring the first number in the larger box down below.
  • Multiply this number by the value in the smaller box
  • Write the product below the next number in the larger box
  • Add vertically, and write this outside of the box
  • Repeat these steps until there are no more operations left to do.


Here are the above steps worked out:

Step 1: Bring the first number down

Steps 2 & 3: Multiply by the number in the smaller box, write this under the next number in the larger box

Step 4: Add vertically, writing this outside the box

We repeat this process to complete the table:

The numbers outside of the box are coefficients to the quotient.  The last number represents the remainder. 

The solution is 2 x squared minus 3 x plus 4 with a remainder of zero.


If there is a non-zero number at the end, then this is the remainder.  To write the remainder in our solution, we must use that number as the numerator of a fraction, with the divisor as the denominator.  For example, let's say we worked through the synthetic division, and our last number was a –2, instead of 0.  We would write our solution as:

2 x squared minus 3 x plus 4 minus fraction numerator 2 over denominator x plus 3 end fraction