Amy has a master's degree in secondary education and has taught math at a public charter high school.
In math, when you read or hear about dividing functions, they are usually talking about dividing polynomials. Recall that polynomials are functions of the following form:
When you divide two such functions together, you get what is called a rational expression. A rational expression is the division of two polynomials. If they divide evenly, your answer will become a polynomial.
I will show you how long division works for dividing polynomials. You can't divide a polynomial with another polynomial whose degree or exponent is larger. Long division of functions uses a very similar process to long division of numbers, as we will see. It uses a circular pattern of comparing, multiplying, subtracting, and carrying down. Let's see how it works by dividing function f by function g.
Remember, our function includes all the terms and not just the first term. So function f(x) is not just x^4, but x^4+3x^2+x+9. Likewise our function g(x) is not just x^2, but x^2+1.
Setting Up the Problem
First, we need to set up the problem for long division. Remember from regular long division that the top number goes inside the division bracket. In our case, the top number is our function f(x). When we set up the functions though, we need to add in our zero values. For numbers, when we have a zero value, we have a zero in its place.
For example, 101 has a zero in the tens place because it doesn't have any tens. Look at our g function. It doesn't have an x value, so we need to add in a zero for that zero value. For our f function, it doesn't have an x^3 value, so we have to add in a zero for that as well. Our long division properly set up looks like this:
Look at the zero values now. Do you see how we put in the zeros where our polynomial didn't have anything there? For the x^2+1 function, did you notice that we didn't have an x value? Because we didn't, we need to put in a 0x as a placeholder just like we do with numbers. For the number one hundred and one we write it out as 101 and not 11. We put in the zero as a placeholder for the tens place even though the number 101 does not have a tens value. It's a similar thing when it comes to dividing polynomials.
Before we begin the long division process, I want to point out to you one difference between dividing functions and dividing numbers. When dividing numbers, you generally look at and compare all the digits in the numbers. This is not so with functions. It's a bit easier with functions actually as you are only concerned about the first terms at every step. Even though we concern ourselves with only the first terms at every step, as we go along, we will have taken care of all the terms by the time we are done.
The Long Division Process
We've added our zero values where they need to go. Now we can go ahead and perform the long division. First, I compare the first terms in each function, the x^2 with the x^4. I ask myself, what do I need to multiply x^2 with to get x^4. My answer is x^2, and so I will write that on top of the x^4. I then go ahead and multiply my g function by x^2 and write the result underneath.
Now, I will subtract the result from my f function inside the division bracket:
My remainder at this point is 2x^2. I'm not done yet. My g function has three terms and my remainder only has one term, so I will bring the next two terms down. I will then compare the x^2 to the 2x^2 and ask myself what do I need to multiply the x^2 with to get to 2x^2. I need a 2. I write that number on the top line on top of the x^2 terms because that is where I am in my long division.
If we had a value for the x^3 position after subtracting, I would be comparing the x^2 with that term instead of the x^2 term, and I would be placing a value on top of the x^3 position. But since our x^3 is 0, we are at the x^2 position.
I will then multiply that value I got from comparing the x^2 with the 2x^2, the 2 with my g function, and write that on a new line. I will then subtract the bottom line from the line on top of it like I do with regular long division. I will see what I get as a remainder.
I will keep repeating the whole process until the remainder has an x with a lower exponent than the first term of my g function. Here is how the problem looks now:
I stopped right here because my remainder x+7 has an exponent of 1, which is lower than the exponent of 2 in my g function of x^2+1. Notice how in my answer line I wrote a +2 because the 2 is positive. Because I am dealing with polynomials, I need to separate the answer terms with either a '+' or a '-' for either positive or negative values, respectively.
Remember from long division that when writing out the answer, the remainder is written over the divisor, in our case, the g function. So, our full answer becomes this:
Polynomials are functions that follow this form:
A rational expression is the division of two polynomials. You can divide two functions that are polynomials. When written in fraction form, the expression becomes a rational expression. Long division is used to divide two polynomials. The procedure is similar to that of numbers.
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