Remember being a kid and going through the anxiety of learning long division? There were only two roads that you took. You either understood it entirely, or you struggled. And now you are in algebra, and you have to perform long division on polynomials.
Polynomial long division is the process of dividing one polynomial into another polynomial, using the same basic techniques as regular long division. The main requirement is that the divisor must be of the same degree or smaller than the polynomial that you divide into.
Let’s break down the polynomial long division process so that you can see how it’s similar to regular long division and what the process looks like, step by step.
Breaking long division down.
Before we jump into long division with polynomials, we will take a look at how regular long division works. We’ll take a look at the different working parts and how we work with numbers as we solve a division problem.
When working with numbers, the parts of a division problem are the divisor, dividend, and quotient. The divisor is the number dividing into the dividend. The dividend is the number that we divide into. The quotient is the result of our division. It tells us how many times the divisor divides into the dividend. It’s our answer to the division problem.
Now let’s go through the steps of regular long division. The first step is to find the number of times the divisor goes into the dividend. Once we figure out this number, we place it above and multiply it times our divisor. We place the result underneath the dividend, and we subtract. We continue dividing, multiplying, and subtracting until all parts of the dividend are processed. Our answer, in the end, is the quotient and the remainder. The quotient tells us how many times the divisor divides into the dividend, and the remainder is what is leftover. Here is an example of the process.
Before diving into performing long division for polynomials, we will look at its basic structure.
The difference that we face in polynomial division is that we use polynomials instead of numbers. The degree of the divisor must be the same degree as the dividend or lower. Once we have our long division problem set up, we can move on to the process itself.
As you work through each step, try to find a way to remember which step you are on in the process. One of the most significant issues most students have when performing long division with polynomials is losing track of where they are in the process. But remember, the more you practice this process, the easier it will become.
Now we go over the steps of long division.
Working through long division, step by step.
Now, let’s look at how long division for polynomials is performed. The process won’t be too complicated if you are familiar with regular long division. The biggest issue I see that people have is in keeping everything organized.
Before we begin, let’s look at the example problem that 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 right 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
I have included a brief video of all the slides explaining the process.
Example long division problems.
In this next section, I will go over two example problems that show how long division works a bit further.
The first problem is x3+x2-33x+63 / x-3.
If we go through and divide our divisor into our dividend, we’ll see that our final result is x2+4x-21. We also have a remainder of 0, which indicates that our divisor is a factor of the original f(x).
This next problem works with a polynomial with a higher degree. The process is still the same. As we work step by step, we see that the result of this division is -3x3 +2x2+27x+18. Our remainder in this problem is also zero.
I know that long division is not easy. Always remember to practice, practice, practice. The more you practice, the easier it gets.😄