Long division and synthetic division are staples in algebra. Some people prefer one over the other. Others would prefer to not use either.
Both long division and synthetic division provide a way of dividing into polynomials. The difference is that long division allows you to divide a polynomial into another polynomial. Synthetic division, the shortcut version of long division, allows you to divide a binomial into a polynomial.
The above explanation is an oversimplified version of what each provides. Let’s dive a little deeper into each one to see where it is most useful in algebra.
What is long division?
Long division is the process of dividing one polynomial into another polynomial. There are many algebraic instances where this process is performed. One such instance is when finding the oblique asymptote of a rational function.
When finding the oblique asymptotes of a rational function, you divide the denominator into the numerator.
Long division is used when the divisor’s degree is less than or equal to the degree of the dividend.
Here is the process for performing long division.
Let’s look at the example problem we will walk through, step-by-step.
Our divisor is equal to x-1. Our dividend is equal to x3+2x2+7x-10.
When we divide our divisor (x-1) into our dividend (x3+2x2+7x-10), the first step we will concentrate on is finding the highest power in the divisor and the highest power in the dividend.
The highest power in the divisor is the x. The highest power in the dividend is the x3.
We then want to find the number of times x divides into x3. In our case, this results in x2.
We then multiply x2 times x-1 and place the result under the (x3+2x2).
We then subtract the (x3-x2)from the (x3+2x2) and place the results under the line. We bring down the 7x and continue the process.
Now we move on to dividing our x into the 3x2.
We then repeat the process. We find the number of times x divides into 3x2 and place it above. We come up with 3x. We then multiply the 3x times (x-1). The result goes under the (3x2+7).
We then subtract the (3x2-3x) from the (3x2+7x). We then drop down the -10 from above and continue. We place the results under the line.
We then repeat the process. We find the number of times x divides into 10x and place it above. We come up with 10. We then multiply the 10 times (x-1). The result goes under the (10x-10).
We then subtract the 10x-10 from the 10x-10. We place the results under the line. Our remainder is zero. The answer indicates that x-1 is a factor of x3+2x2+7x-10.
Whenever we get a remainder that is zero in long division, it indicates that our divisor is a factor of the polynomial that we are dividing into. What this means is that:
- (x-1) (x2+3x+10) = x3+2x2+7x-10
What is synthetic division?
Synthetic division is the process of taking a binomial and dividing it into a polynomial. Whereas long division allows a polynomial to be divided by a polynomial, synthetic division only allows a polynomial to be divided by a binomial.
Also, unlike long division, it is the shorter version of polynomial division.
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, 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 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.
Is it better to use long division or synthetic division?
Most people think long division and synthetic division can be used interchangeably, but that isn’t true.
One thing to remember is that the best method to use depends on two things.
- If you are dividing a binomial or a polynomial into a polynomial
- The output you wish to see once your calculations are complete.
If you are dividing a binomial into a polynomial, either long division or synthetic division can be used. But if you are dividing a polynomial into another polynomial, the best method to use is long division.
There is only one thing you have to watch out for in long division; the degree of the divisor must be equal to or less than the degree of the dividend.
If you still aren’t sure which method to use, think about how your output should come out. If you are seeking a binomial instead of a single numerical answer, it would be better to use long division to find your solution.
For instance, when trying to find the oblique asymptotes of a rational function, it would be better to use long division for your calculations. The output should be the equation of a line, and that’s the output you will get when performing long division.
Because of the convenience of synthetic division, I tend to use that more than anything else. It’s clean, and it helps me stay organized through the process. But as long as you remember the first constraint, you should be able to pick a method that works for you.
Here are a few examples.
I will break down two examples of problems where only long division works best and then another where synthetic division works.
The first example involves finding the oblique asymptote of a rational function. To find the asymptote, you must divide the denominator into the numerator. The result should be the equation of a line that represents the oblique asymptote.
We will look at the equation f(x)=x3/x2+1.
I love convenience, but there are two conditions we should consider before choosing synthetic division over long division in this example.
The first is that the result we are seeking is the equation of a line. Since long division preserves the answer as an equation, it’s easier to use long division over synthetic in this case.
The second condition is that we are dividing a polynomial by a polynomial. Synthetic division only allows for a polynomial to be divided by a binomial. In this case, we are stuck with long division.
The equation for the oblique asymptote is f(x)=x.
In the second example, we will evaluate the function f(x)=x5-3x4+4x3-6x2+5x-1 at f(2).
Most people would substitute 2 into this equation and solve. When we use the remainder theorem along with synthetic division, we take the hassle out of trying to solve higher-degree polynomials for different values.
Using synthetic division is much more convenient in this situation. In this case, f(2) = 1.
There are many instances where it will be confusing when it is appropriate to use either technique. Once you do more exercises, picking which technique to use will be a piece of cake.