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Simplifying Rational Expressions With Factoring

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will review rational expressions and factoring. Then, we will see how to use factoring to simplify rational expressions and actually put the process to use in an example.

Rational Expressions

Suppose you are in a work meeting working on a problem for your boss. You and a coworker are working together on a problem of finding an expression to represent some data that are going to be used in maximizing profit for the company. You come up with a solution of (2x + 4) / 2x. At the same time, your coworker says they found a solution of (x + 2) / x. Which one of you is right?


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As it turns out, you both are, and it has to do with simplifying rational expressions using factoring. There are two things involved with simplifying rational expressions using factoring - rational expressions and factoring! So, before getting to the actual process, let's first make sure we are familiar with both of these concepts, starting with rational expressions.

In mathematics, a rational expression is basically an algebraic expression divided by an algebraic expression. For example, both of the solutions that you and your coworker came up with are rational expressions because they both have an algebraic expression divided by an algebraic expression. In the image, we see a few more examples of rational expressions.


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Well, that's pretty straightforward. Now, let's take a look at factoring.

Factoring

You may be familiar with the fact that when we multiply two things together to find their product, we call those things factors. Sometimes, we are given a product, and we want to find the two factors that we needed to multiply together to get that product. This is called factoring.

In simplest terms, consider factoring a number. For instance, consider the number 6. To factor the number 6, we simply want to figure out what numbers we multiply together to get 6. Well, we can multiply 1*6 to get 6, or we could multiply 2*3 to get 6. Both of these represent a factorization of 6.

Similarly, we can factor algebraic expressions. For instance, consider the numerator of your work solution, 2x + 4. This is actually a product of two factors. To find these factors, we are going to recognize that 2x + 4 can also be written as 2x + 2*2. Notice that both terms have a factor of 2 in them. We can actually pull that two out to get 2(x + 2). In doing this, we have factored 2x + 4 to get 2(x + 2).

There are many different ways to factor algebraic expressions, and those processes each could be a lesson in themselves. Therefore, in this lesson, we will stick to simple examples involving factoring, and the factorizations of expressions will be given in different examples. Now, let's get to the good stuff! Time to look at simplifying rational expressions using factoring.

Simplifying Rational Expressions Using Factoring

Okay, we know what rational expressions are, and we know what factoring is, so let's get to it! You are probably familiar with simplifying fractions. For example, the fraction 2/4 can be simplified to 1/2, because we can rewrite 2/4 as 2*1 / 2*2, and then cancel out the common factor of 2 that is in the numerator and denominator.


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That's not so hard, is it? Well, I've got good news! Simplifying rational expressions is the exact same process as simplifying fractions, so there's no need to be intimidated by it! When simplifying rational expressions using factoring, we use the following steps:

  1. Factor the numerator and denominator as much as possible.
  2. Cancel out any factors that are in both the numerator and denominator. The result is your simplified expression.

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