When studying higher level mathematics, such as calculus and algebra, synthetic division can help simplify quite complex divisions and processes. This is a valuable tool to help speed up the process and get to the desired answer more quickly. However, when performing a synthetic division problem, you must be able to correctly calculate the remainder for the problem for it to be accurate. In this article, we will discuss what synthetic division is, how to perform a synthetic division problem, working through an example of a synthetic division problem, finding the remainder in a synthetic division problem, tips for solving synthetic division problems and common mistakes to avoid when performing synthetic division.

What is Synthetic Division?

Synthetic division is a method of simplifying complex polynomial long division problems. It is a simplified method of the traditional long division process and can often give you the desired result more quickly and easily than using the standard long division process. It is useful for solving polynomials with a maximum degree (the largest exponent sum of all its terms) of 3 or 4. Occasionally, it can also solve polynomials with a degree of 5 if the highest degree terms are simple.

How to Perform a Synthetic Division Problem

Synthetic division follows a set structure to ensure accuracy. It requires the following steps:

  • Write down the polynomial you want to divide in typical polynomial form, with the highest degree (or “leading”) term first and each lower degree term following consecutively.
  • Write the divisor (either an integer or polynomial) to the right of the polynomial, separated by two vertical lines.
  • Write 0’s in every row below the coefficients from left to right.
  • Divide each coefficient of the dividend by the divisor. Then for each row below it, multiply it by the divisor and add it to the next coefficient.
  • Continue this process until all rows in this table are filled out. The number at the bottom is your remainder.

Working Through an Example Synthetic Division Problem

Now let’s look at an example problem to help demonstrate how this works. Say you want to divide x³-3x²+3x+7 by x-2. Here are the steps you should take:

  1. Write x³-3x²+3x+7 by its highest degree to lowest degree terms: x³-3x²+3x+7. Also write the divisor x-2 separated by two vertical lines.
  2. Fill in the empty rows below each coefficient with 0’s, starting from left to right.
  3. Divide the first coefficient (1) by the first coefficient of the divisor (1). This equals 1.
  4. Multiply the 1 in the divisor column by -3 and add it to the -3 in the second row below it (1 x -3 = -3 -3 = -6). Then divide this by the first coefficient of the divisor (1). This equals -6.
  5. Multiply the 1 in the divisor column by 3 and add it to the 3 in the third row below it (-6 x 1 = -6 + 3 = -3). Divide this by 1. This equals -3.
  6. Multiply the 1 in the divisor column by 7 and add it to 0 in in fourth row below it (-3 x 1 = -3 + 7 = 4). Divide this by 1. This equals 4.

The result of this synthetic division problem would be (x²-6x-3) + 4. The nice thing about synthetic division is that you don’t have to spend much time writing down long division steps as you might have with traditional long division—it’s all compressed into one step. The 4 is your remainder for this problem and can either be left in fraction form or converted into a decimal if desired.

Finding the Remainder in a Synthetic Division Problem

Now that we have an idea of how a synthetic division problem works, let’s look at how you can find the remainder specifically. Essentially, Synthetic Division works by expanding out each of the coefficients and numbers that make up the dividend, then breaking up each coefficient into polynomials spanning certain degrees until you reach your remainer. To find your remainder in a Synthetic Division problem:

  • Write down your dividend and divisor with two vertical lines between them.
  • Start with the last number in your dividend and divide by the first number in your divisor. Whatever your result is, that number is your remainder.

Tips for Solving Synthetic Division Problems

Synthetic division can help speed up complex polynomial long division processes, but it’s important to understand all of its steps before jumping into a problem. Here are some helpful tips for solving synthetic division problems:

  • Start with simple practice examples before jumping straight into complex problems.
  • Always double check your work to make sure no mistakes were made while breaking up coefficients or multiplying numbers.
  • Draw out visual diagrams if needed to help break up coefficients into polynomials more easily.
  • When solving more complex problems, use a calculator if available for extra accuracy.

Common Mistakes to Avoid When Performing Synthetic Division

It’s important to be mindful of potential mistakes that may occur when performing synthetic divisions. Here are some common mistakes to avoid when performing synthetic division:

  • Not breaking down coefficients into more specific degrees before performing synthetic division.
  • Failing to accurately multiply numbers while filling out rows.
  • Not verifying your answer after completing synthesis division—always double check your work!

Resources for Further Learning About Synthetic Division

There are many online resources available to help you further understand how synthetic division works and practice performing these type of problems. Here are some great resources you can use:

By understanding how Synthetic Division works and avoiding common mistakes, you can easily figure out how to find remainders in these problems and solve them accurately and quickly in no time!