How to Simplify Rational Expressions
Simplifying Rational Polynomial Functions
One thing you may counter when you’re working with polynomial functions is a situation like this. We have two functions that are defined for us, g(x) and h(x). Then, we’re told that we have a third function f(x) that is defined based on the other two functions and is one of those functions divided by the other of those functions.
We’re asked to simplify this function into the simplest terms possible. Let’s look at an example of how to solve this. Let’s say for our two defined functions we have x squared plus x minus 6, and x squared plus 5x minus 14. The first thing we want to do is we want a rewrite f(x) in terms of what we know.
We have f(x) is equal to h(x) on top. x squared plus 5x minus 14 divided by x squared plus x minus 6. What we want to do from this point is we want to factor both of these polynomials to make sure that if there are any roots that the two have in common that we can cancel it.
We’re going to assume that we can factor them and see what we can get. For this first one, we have -14 as our product of the two numbers we’re looking for. 14 can be factored as 1 and 14 or 2 and 7. One of the two numbers has to be negative. What we’re looking for is 7 minus 2, because 5 is equal to 7 minus 2.
We have x plus 7 and x minus 2. For the second polynomial, we have 6 as our product of the two factors here. 6 can be factored as 1 and 6 or 2 and 3. Since it’s negative, we need a positive and a negative. What we’re looking for is 3 and minus 2. x plus 3 and x minus 2.
You’ll notice from our factorization here we have an x minus 2 term on the top and the bottom. When we go to simplify this, what we’re going to do is we’re going to cancel the x minus 2 term, and we can rewrite this function, f(x), as x plus 7 divided by x plus 3.
It is very important to note that this function is undefined at x equals 2, because x equals 2 causes the denominator to be 0. Just because we’ve eliminated the x minus 2 factor when we simplified, based on how the function is defined, the function is still undefined at x equals 2. It’s defined as the ratio of these two functions.
This function is undefined at x equals 2. When we rewrite the simplification of this function, we have to write it as f(x) equals x plus 7 divided by x plus, where x does not equal 2. It’s very important to include this. x is also not able to equal 3, because that also makes the function undefined.
This is an example of how to simplify an expression where it’s a ratio of two polynomials. As I mentioned several times already, but I’m going to repeat one more time, it’s very important to always include the fact that the function will be undefined wherever the denominator function is equal to 0, regardless of whether you factored it out or not, or eliminated it from the simplification or not. This is the process for doing that.