Using Synthetic Division to Save Time in Calculus and Algebra

Student in a library, wearing headphones and studying — Once you get to the point in math where you need to start dividing polynomials, knowing how to use the synthetic division method will save you a lot of time. SDI Productions / Getty Images

In algebra and calculus, a polynomial function is used to chart out graphs and waves with much more complexity than a simple linear factor. Polynomial division is sometimes required to factor them, and cut them up into chunks that we humans can better understand.

That's where the synthetic division method comes in. Old-fashioned long division can also be used to factor polynomials, but the synthetic method is often easier and faster.

Synthetic Division vs. Long Division

Long Division

You may be familiar with long division from grade school, so-named because it can be a hassle that takes forever to get results.

To perform long division, you must multiply your divisor by a quotient number, then subtract the result from your dividend. Then you repeat this step until you end up with a remainder which is smaller than the original divisor. If it divides evenly, you get zero remainder.

The long division method works if you want to, say, find out how many times 15 fits into 83. It's good for whole numbers and very simple fractions, but when polynomial division gets thrown into the mix, long division immediately becomes much confusing and time consuming.

Synthetic Division

The synthetic division method simplifies this process by switching up the visual framework in which division is applied. Ultimately, we are trying to get the same result as long division, but if we use synthetic division, it arranges all the pieces we need into a neat grid that is easier to read.

A synthetic division calculator can also help you work through this process.

Linear Factor and Polynomials

The simplest form of a polynomial is known as a linear factor, also sometimes called a binomial. In school, you may recall seeing linear factors in the pervasive y = mx + b function format. When it comes breaking down a more complex polynomial such as a quadratic function, we must take our whole polynomial and divide it by a linear factor.

Ultimately, you may repeat this process until only linear factors are left in the solution of the equation. It's similar to breaking down a whole number like 12 into chunks (2 x 3 x 2) which make up the whole when multiplied together. However, just like like some whole numbers, polynomials are not necessarily made to be broken up so perfectly.

Using Synthetic Division to Factor a Polynomial

1. Find the Degree of the Polynomial

The degree of a polynomial is equal to the greatest power present in the function. For instance, if 2x³ is the largest power number present in the function, then that polynomial will be considered to have a degree of three.

This value is important because it tells you how many times the polynomial will have to be divided to be completely factored out.

2. Sort by Leading Coefficient

The leading coefficient of a polynomial is the variable of the function attached to the highest power. To perform synthetic division, you first must sort the polynomial by highest coefficients in descending order.

For any coefficient that doesn't have a corresponding power, insert a zero value. This will be helpful for later steps.

As an example, the polynomial x² + 6 + 3x⁴ + 8x becomes 3x⁴ + 0x³ + x² + 8x + 6 when sorted by leading coefficient.

Now, we our going to take our sorted polynomial and divide it by the linear factor x + 1 using synthetic division. Screenshots in this guide were taken using E Math Help's synthetic division calculator.

3. Performing Synthetic Division

Take the constant (1) from the divisor (x + 1) and multiply it by -1. Then add it to the left side in a small box or a separated area. This is the number you’ll use for the synthetic division.
Bring down the leading coefficient next to the box.
Multiply the number in the box by the number you just brought down and write the result underneath the second coefficient.
Add the numbers in the second column and write the sum underneath.
Move over to the third column. Repeat the multiplication and addition steps until you’ve gone through all the coefficients. This is the basis of the synthetic division method.
The last number you get is the remainder, and the other numbers form the coefficients of the quotient polynomial. If the polynomial divides perfectly into the linear factor, you'll end up with zero remainder.

The result will give you the quotient and the remainder of the division. The degree of the quotient polynomial will be one less than the degree of the original polynomial — and just like that, we've performed polynomial division.

After we use synthetic division to factor the polynomial 3x⁴ + x² + 8x + 6 divided by (x + 1), our final answer is a lower degree polynomial of 3x³ – 3x² + 4x + 4 and a remainder of 2 divided by (x + 1).

The result probably looks similar to what you would expect using the long division method, but hopefully you find this new method to keep the data much more organized.

Who Invented Synthetic Division?

Synthetic division is widely accepted to have been developed by the 18th century mathematician Paulo Ruffini (1765-1822). Ruffini had first applied this method to divide polynomials which start with a coefficient of one, which he called Ruffini's rule.

Some time later, other mathematicians figured out that using synthetic division could help factor all kinds of polynomials. Referring to it as Ruffini's rule also fell by the wayside as it took on the much more clinical name of synthetic division.