## 8: Rational Expressions - Mathematics

**College Algebra**

**Tutorial 8:** **Simplifying Rational Expressions**

**A rational expression is one that**

**can be written in the form**

**where P and Q are polynomials and Q does not equal 0.**

Domain of a Rational Expression |

So, when looking for the domain of a given rational function, we use a back door approach. **We find the values that we cannot use, which would be values that make the denominator 0.**

**Example 1** : Find all numbers that must be excluded from the domain of .

So to find what values we need to exclude, think of what value(s) of *x* , if any, would cause the denominator to be 0.

Since 1 would make the first factor in the denominator 0, then **1 would have to be excluded.**

Since - 4 would make the second factor in the denominator 0, then **- 4 would also have to be excluded.**

**For any rational expression , and any polynomial R, where ,, then**

This will come in handy when we simplify rational expressions, which is coming up next.

Simplifying (or reducing) a Rational Expression |

**Example 2:** Simplify and find all numbers that must be excluded from the domain of the simplified rational expression: .

***Divide out the common factor of ( x + 3)**

***Rational expression simplified**

Looking at the denominator *x* - 9, I would say it would have to be *x* = 9. Don’t you agree?

**9 would be our excluded value.**

**Example 3:** Simplify and find all numbers that must be excluded from the domain of the simplified rational expression: .

***Factor out a -1 from (5 - x )**

***Divide out the common factor of ( x - 5)**

***Rational expression simplified**

To find the value(s) needed to be excluded from the domain, we need to ask ourselves, what value(s) of *x* would cause our denominator to be 0?

Looking at the denominator *x* - 5, I would say it would have to be *x* = 5. Don’t you agree?

**5 would be our excluded value.**

To get the most out of these, **you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem**. At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problem 1a:Find all numbers that must be excluded from the domain of the given rational expression.

Practice Problems 2a - 2b:Simplify and find all numbers that must be excluded from the domain of the simplified rational expression.

**Need Extra Help on these Topics?**

**http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/****int_alg_tut32_multrat.htm**

The beginning of this webpage goes through how to simplify a rational expression.

Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.

Last revised on Dec. 14, 2009 by Kim Seward.

All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.

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## Multiplying Rational Expressions

A rational expression is a fraction in which either the numerator, or the denominator, or both the numerator and the denominator are algebraic expressions.

When two fractions are multiplied, we multiply the numerators of the fractions to form the new numerator and we do the same for the denominators. This is the same with rational expressions. If there are common factors in both numerator and denominator of the two rational expressions then we may cancel them before we multiply.

Simplify the following expressions:

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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## Rational Expressions

Hi, and welcome to this video about **Rational Expressions**.

Before we talk about what rational expressions are and the operations that can be performed with them, it may be a good idea to review some terminology. A **polynomial** is a group of algebraic or numeric terms that are joined by the operations of addition or subtraction. There are different types of polynomials based on the number of terms that are present:

A **rational expression** is nothing more than a ratio of polynomials. As you know from previous practice with **ratios**, you cannot divide by 0. It is important to keep this in mind when dealing with rational expressions because allowing a value of 0 in the denominator would create an expression that is “undefined.”

Using function notation for polynomials (p(x)) and (g(x)), a rational expression can be defined like this:

This example shows a rational expression with a monomial, (5x), in the numerator and a binomial, ((x-2)), in the denominator. The value (x=2) is the excluded value, as it would result in a denominator of 0. This expression cannot be simplified further.

**Addition and Subtraction of Rational Expressions:**

Rational expressions cannot be added or subtracted unless they share a common denominator. Algebraic rules allow us to adjust fractions to create common denominators as long as we make the same adjustment to the numerator. Let’s look at an example with fractions:

In order to add (frac<1> <5>+ frac<3><7>), we must create a common denominator. Specifically, we need to determine the **least common denominator**, meaning the smallest multiple of 5 and 7. In this case, that number is 35. The adjustment to each fraction that needs to be made to create the common denominator is:

We need to multiply the first denominator of the first expression by 7 to get to 35, but we also must multiply the numerator by the same value. Because we have simply created an equivalent fraction to allow us to add. Likewise, the second expression must be multiplied by (frac<5><5>), in order to create 35 in the denominator. After these adjustments are made and the denominators are the same, simplify the numerators:

Rational expressions are added and subtracted the same way. Typically, the expressions need to be factored before the least common denominator can be determined and domain restrictions (excluded values) should be noted. Consider this example:

Now we want to determine the lowest common denominator. What is the smallest multiple of ((x-2)) and ((x-2)(x+1))?

Alright now that we have our equations written out, we want to make sure that we don’t have any domains that need to be excluded. Which, we do. Remember we don’t want 0 in the denominator position. So, in these scenarios we know that (x eq 2), or over here, -1. If (x=-1) this would end up being 0, multiplied by another term, still remains 0. The 0 in the denominator, we can’t have that. Over here, if (x=2), (2-2=0), again, we can’t have a 0 in the denominator, so these are our two terms, our domains that need to be excluded.

Alright now we need to adjust the first expression by multiplying by the factor needed to match the least common denominator. So if we want our first term here, to match this term over here in the denominator position, we’re going to multiply by (x+1) in the numerator and the denominator.

Now we’re going to rewrite the expression as a fraction, and simplify the numerator. And now we have our answer:

Now let’s move on to **Multiplication and Division of Rational Expressions**:

Here are the three steps to multiplying rational expressions. Now remember, when multiplying fractions, numerators and denominators are multiplied straight across.

Step #1: Factor the numerator and denominator of each expression being multiplied.

Step #2: Simplify by canceling out common factors from the numerator and the denominator.

Step #3: The final answer is what is left after canceling. You may be asked to include domain restrictions with your solution.

Let’s use these steps to solve an example problem:

Now, because we have like terms in the numerator and the denominator position, we’re able to cancel them out. That leaves us with:

But we can simplify this even further, remember, 6 is a **factor**of 36, so let’s simplify:

And yet, we can simplify this again, remember, you have an *x* in the numerator and an *x* in the denominator, so let’s simplify:

And now we have our answer, (frac

Dividing rational expressions includes one extra step at the beginning of the process. When dividing by a fraction, it is the same as multiplying by the reciprocal of the second fraction. You can remember this rule as, “Keep, Change, and Flip” which translates to keep the first fraction, change the operation to multiplication, and take the reciprocal (or flip) of the second fraction.

Keep in mind that domain restrictions must be considered from both the numerator and denominator of the second fraction because of the “flip” in the division process.

So here’s our problem, now remember our three steps, *keep* the first fraction, *change* the operation, and then, *flip*. So here we go:

Here is now where we multiply, cause we kept the first fraction, we changed to multiplication, and then we flipped the fraction over here. So, time to multiply.

So now we have our answer: (frac<3x^<2>><4(x+6)>)

But remember, that’s not our complete answer if we don’t include our restricted domain, we have (x eq 0), and (x eq -6). Remember we have to make sure that we don’t have a 0 in the denominator or the numerator of our second term.

## Reducing Rational Expressions – Polynomials EVERYWHERE!

Hey guys! Welcome to this video on **Simplifying Rational Expressions**.

A **rational expression** just refers to a **fraction** with a polynomial in the numerator, and a polynomial in the denominator.

One thing that we need to keep in mind when working with rational expression is that divisibility by 0 is not allowed. Just like when dealing with regular numbers, you cannot divide by 0. So, when dealing with a rational expression, we always assume that whatever *x* is, it will not give us division by 0.

Alright let’s take a look at how to reduce a rational expression. We’re actually doing the same thing we would do when reducing a regular fraction.

So, let’s say we have (frac<18><8>). When we reduce this, we can cancel our like terms. So we can rewrite this as:

We can cancel our 2s here giving us:

So now we have a fraction reduced down to its simplest form. There is not another number that both our numerator and denominator are divisible by.

It works the same way with a rational expression.

Let’s try reducing our first example.

We can rewrite our numerator, once we factor this out, as:

And once we do this, we can see that our ((x+4))s will cancel out. So we cancel that out, leaving us with:

Now, we need to be careful when canceling terms. The only reason we were able to cancel out our ((x+4))s here was because they are both being multiplied in the numerator and the denominator. This would not work if our top was: (frac<(x+4)+(x-4)><(x+4)>).

Let’s now move on to our second example, which is a bit trickier.

We can do the same thing that we did in our first example by rewriting our numerator and denominator. So that would give us:

So, we can go ahead here and cancel our ((x-4))s, which would leave us with:

For our last example we have:

To reduce it, we can rewrite our numerator by factoring out a 4. Which would give us (4(x+1)). In the denominator we can factor out an (x^<2>), which would give us (x^<2>(x^<2>-1)).

But, notice, we can factor this out even further so we can get something to cancel out with our numerator here.

At this point, we can cancel out our ((x+1))s here, leaving us with:

There is no further reduction we can do, so we now have it in our simplest form.

I hope that this video has been helpful for you. For further help, be sure to check out more of our videos by subscribing to our channel below.

## 8 is my lucky number

In Algebra 2, we are learning how to simplify, add, subtract, multiply, and divide rational expressions. They always need a good day of practice to let it sink in.

First, I started them off working in partners with 5 row game examples. They have different problems that should lead to the same answer. If they don't get the same answer, they switch papers and find the other person's mistake. They did well with this.

Row Games - Rational Expressions

Then, they were ready to move onto a whole class competitive activity called Spoons. I was so excited to try this after I read it on This website. It was a lot to create the activity. In total, it is only 6 questions with 6 answers plus I added some incorrect solutions. However, it took some time to get everything ready. I put 4 of the same problem on one page and printed those out and cut them out. Students worked in groups of four, so I clipped them together. I have one class of 18 and one class of 29. I did this yesterday with my small class and it went well. I am a bit nervous to do it in my big class, but I will. So, it was a lot of printing, cutting, and paperclipping for the problems. Then, for the answers, I did 8 of each answer, cut them out, glued them to a big index card, and brought them to the library to get laminated - love laminating! ***Be sure you don't have any mistakes or typos before you laminate. I did, it is a bummer.

I had the kids move into groups and use a little table in the middle to put all the answer cards on, face down and off to work they went. It was competitive, it was crazy, but they had fun. It took about 20 minutes to do the 6 questions.

Oh, and I added this in the directions of the activity, you may want to add a picture of a spoon to the back of the index card before you laminate it. I wish I had. I explained the whole game and we played it and class time was almost up and a student said, "But, wait, we didn't get to play spoons yet." Hmmmm.

By the way, when I was talking to my colleagues about doing this, I was surprised how many people have never heard of the real spoons game. I think we need to play it!

## Simplify Rational Expressions

How to simplify rational expressions? A tutorial with examples and detailled solutions is presented.

To simplify a rational expression, we first factor both the numerator and denominator completely then reduce the expression by cancelling common factors.

** Example 1:** Simplify the rational expression

__Detailed Solution to Example 1__

- Factor both the numerator and denominator completely.

** Example 2:** Simplify the rational expression

__Detailed Solution to Example 2__

- Factor both the numerator and denominator completely.

** Example 3:** Simplify the rational expression

__Detailed Solution to Example 3__

- Factor both the numerator and denominator completely.

## 8: Rational Expressions - Mathematics

Some of the content of this guide was modeled after a guide originally created by the **Openstax** and has been adapted for the GPRC Learning Commons in September 2020. This work is licensed under a Creative Commons BY 4.0 International License.

**Rational Expressions and Non-Permissible Values:**

##### A rational expression is the ratio of two polynomials:

**Non-Permissible Values:**

##### Non-permissible values are values of the variable(s) that make the denominator of a rational expression equal to zero. In order to find the non-permissible values of a rational expression one has to set the denominator equal to zero and solve it for the variable. Below, we look at few examples on how to find non-permissible values.

##### Example: Find non-permissible values of .

##### We set the denominator equal to zero:

.

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