Back To Course

Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 55,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Kathryn Maloney*

Kathryn teaches college math. She holds a master's degree in Learning and Technology.

Adding and subtracting rational expressions can feel daunting, especially when trying to find a common denominator. Let me show you the process I like to use. I think it will make adding and subtracting rational expressions more enjoyable!

Remember back when we added and subtracted fractions? Well, a rational expression is simply a fraction with *'x*'s and numbers. We follow the same process for adding and subtracting rational expressions with a little twist. Now we may need to factor and FOIL to simplify the expression.

The process we will follow is:

- Factor
- Find the common denominator
- Rewrite fractions using the common denominator
- Put the entire numerator over the common denominator
- Simplify the numerator
- Factor and cancel, if possible
- Write the final answer in simplified form

As we get started, let's also remember that to add or subtract fractions, we need a common denominator. Try this mnemonic to help you remember when you need a common denominator and when you don't:

*Add Subtract Common Denominators, Multiply Divide None.*

*Auntie Sits Counting Diamonds, Mother Does Not.*

Let's look at our first example.

(*x* + 4)/(3*x* - 9) + (*x*- 5)/(6*x*- 18)

First, we need to factor.

(3*x* - 9) = 3 (*x*- 3) and (6*x* - 18) = 6 (*x* - 3)

After we replace the factored terms, our new expression looks like:

(*x* + 4)/3 (*x* - 3) + (*x* - 5)/6 (*x* - 3)

To find our common denominator, we simply write down our denominators. From the first term we have 3 (*x* - 3) as our denominator. We write that down for our common denominator. When we look at the second expression's denominator, 6 (*x* - 3), we notice that 6 = 3 * 2. So the second expression has 2 * 3 (*x*- 3). We already have 3 (*x* - 3) written, so the only piece not used is 2. We write that down multiplied by 3 (*x* - 3). Our common denominator will be 2 * 3 (*x* - 3) or 6 (*x* - 3).

Our next step is to multiply each piece of the expression so we have 6 (*x* - 3) as our new denominator. In our first fraction, we need to multiply by 2 over 2. This will give me 2 (*x* + 4)/2 * 3(*x* - 3). Looking at the second fraction, I notice I already have 6 (*x* - 3) in the denominator, so I can leave this one alone.

Now let's write the entire numerator over our common denominator:

2(*x* + 4) + (x - 5)/6(*x* - 3)

Let's simplify the numerator.

2(*x* + 4) = 2*x* + 8

2*x* + 8 + (x - 5)/6(*x* - 3)

Collect like terms in the numerator.

3*x* + 3/6(*x* - 3)

Factor the numerator if possible.

3*x* + 3 = 3 (*x* + 1)

The 3 over 6 reduces to 1 over 2. There isn't anything to slash or cancel, so we distribute in the numerator and denominator for our final answer:

*x* + 1/2*x* - 6

(*x* - 2)/(*x* + 5) + (*x*^2 + 5*x* + 6)/(*x*^2 + 8*x* + 15)

First, we need to factor.

*x*^2 + 5*x* + 6 = (*x* + 3)(*x* + 2)

*x*^2 + 8*x* + 15 = (*x* + 5)(*x* + 3)

After we replace the factored terms, our new expressions looks like:

(*x* - 2)/(*x* + 5) + (*x* + 3)(*x* + 2)/(*x* + 5)(*x* + 3)

To find our common denominator, we simply write down our denominators. From the first term, we have (*x* + 5) as our denominator. In the second term, we have (*x* + 5) and (*x* + 3). Since we already have (*x* + 5) written as part of our common denominator, we will just write (*x* + 3). So, our common denominator is (*x* + 5)(*x* + 3).

Our next step is to multiply each piece of the expression, so we have (*x* + 5)(*x* + 3) as our new denominator. In the first fraction, we need to multiply by (*x* + 3) over (*x* + 3). This will give us (*x* - 2)(*x* + 3)/(*x* + 5)(*x* + 3) as our first fraction. Looking at the second fraction, I notice I already have (*x* + 5)(*x* + 3) in the denominator, so I can leave this one alone.

Now, let's write the entire numerator over our common denominator.

((*x* - 2)(*x* + 3) + (*x* + 3)(*x* + 2))/(*x* + 5)(*x* + 3)

Let's simplify the numerator by writing the numerator over our common denominator and FOIL.

(*x* - 2)(*x* + 3) = (*x*^2 + *x* - 6) and

(*x* + 3)(*x* + 2) = (*x*^2 + 5*x* + 6)

Collect like terms in the numerator.

2*x*^2 + 6*x*

Factor the numerator if possible.

2*x*(*x* + 3)

Our expression now looks like:

2*x*(*x* + 3)/(*x* + 5)(*x* + 3)

We can slash, or cancel, (*x* + 3) over (*x* + 3).

This gives us our final answer, 2*x*/(*x* + 5).

(*x*^2 + 12*x* + 36)/(*x*^2 - *x* - 6) + (x + 1)/(3 - x)

First, we need to factor.

(*x*^2 + 12*x* + 36) = (*x* + 6)(*x* + 6)

(*x*^2 - *x* + 6) = (*x* - 3)(*x* + 2)

After we replace the factored terms, our new expressions looks like:

(*x* + 6)(*x* + 6)/(*x* - 3)(*x* + 2)) + (*x* + 1)/(3 - *x*)

To find our common denominator, we simply write down our denominators. From the first term, we have (*x* - 3)(*x* + 2) as our denominator. In the second term, we have (3 - *x*). I could write (3 - *x*) as part of the common denominator, but I know that -1 * (*x* - 3) = (3 - *x*). So, now it will match with the denominator (*x* - 3).

Now, our expression looks like:

(*x* + 6)(*x* + 6)/(*x* - 3)(*x* + 2)) + (*x* + 1)/-1(*x* - 3

And that -1? It can be put into the numerator. Remember, 1/-1 = -1/1 = -1. It doesn't matter where I put the -1 in the fraction as long as I have a +1 to match it.

So, our common denominator is (*x* - 3)(*x* + 2).

In the first fraction, I already have the common denominator (*x* - 3)(*x* + 2), so I leave that one alone. In the second fraction, I need to multiply by (*x* + 2) over (*x* + 2). This gives us the common denominator of (*x* - 3)(*x* + 2).

Our expression now looks like:

(*x* + 6)(*x* + 6)/(*x* - 3)(*x* + 2) + (-1)(*x* + 1)(*x* + 2)/(*x* - 3)(*x* + 2)

Let's simplify the numerator by writing the numerator over our common denominator and using FOIL, which is *First Outside Inside Last*.

(*x* + 6)(*x* + 6) = *x*^2 + 12*x* + 36

and

(-1)(*x* + 1)(*x* + 2) = (-1)(*x*^2 + 3*x* + 2) = -*x*^2 - 3*x*- 2

Collect like terms in the numerator. Our expression now looks like:

(9*x* + 34)/(*x* - 3)(*x* + 2)

The numerator doesn't factor, so our last step is to FOIL the denominator.

Our final answer is (9*x* + 34)/(*x*^2 - *x* - 6).

The process we follow is:

- Factor
- Find the common denominator
- Rewrite fractions using the common denominator
- Put the entire numerator over the common denominator
- Simplify the numerator
- Factor and cancel, if possible
- Write the final answer in simplified form

Simplifying rational expressions may feel like a daunting process right now, but with practice, you will get better. One tip from me to you: If you don't have the right answer the first time, don't erase the entire expression. Start from the beginning of your work, and look for little mistakes. Many of my students have the right idea, just a misplaced sign or a factoring error.

Once you finish this lesson you'll greatly improve your ability to add and subtract rational expressions.

To unlock this lesson you must be a Study.com Member.

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
4 in chapter 8 of the course:

Back To Course

Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

- Go to Functions

- How to Multiply and Divide Rational Expressions 8:07
- Multiplying and Dividing Rational Expressions: Practice Problems 4:40
- How to Add and Subtract Rational Expressions 8:02
- Practice Adding and Subtracting Rational Expressions 9:12
- Rational Equations: Practice Problems 13:15
- Go to Rational Expressions

- Social & Emotional Development Lesson Plans & Activities
- Foreign Language Lesson Plans & Games
- Famous Children's Authors Lesson Plans
- Animal Taxonomy & Characteristics Lesson Plans
- Physical Education Lesson Plans & Activities
- Famous Athletes Lesson Plans
- Yoga & Flexibility Lesson Plans
- Worm Lesson Plans
- Outdoor Art Lesson Plans & Activities
- Outdoor Safety Lesson Plans
- Teacher Retirement System of Texas Withdrawal
- New York State Science Standards
- Third Grade New York State Science Standards
- Wisconsin State Teaching Standards
- New York State Science Standards for Grade 4
- Arizona English Language Proficiency Standards
- Arizona English Language Proficiency Standards & Levels

- Risk Analysis Plans for Businesses: Techniques & Examples
- Self-Advocacy IEP Goals
- How to Use LinkedIn to Develop Business Relationships
- Power Series: Formula & Examples
- A Cup Of Tea by Katherine Mansfield: Summary & Theme
- Multiplying Integers Games
- The Necklace: Symbolism & Irony
- Benjamin Harrison Lesson for Kids: Biography & Facts
- Quiz & Worksheet - Allergens, Non- & Self-Antigens
- Quiz & Worksheet - Helping Customers During the Selling Process
- Quiz & Worksheet - Law & Performance Reviews
- Quiz & Worksheet - Dimensions of Critical Thinking
- Quiz & Worksheet - Trust Building in Business Teams
- Hypothesis Testing in Statistics Flashcards
- Summarizing Data Flashcards

- TExES History 7-12: Practice and Study Guide
- Introduction to Business: Certificate Program
- ACT Prep: Tutoring Solution
- Introduction to Statistics: Homework Help Resource
- Post-Civil War American History: Homework Help
- Portions of the AP European History Exam: Homework Help
- Punctuation in Writing: Help and Review
- Quiz & Worksheet - Exponentials, Logarithms & the Natural Log
- Quiz & Worksheet - Finding Derivatives of Implicit Functions
- Quiz & Worksheet - Using Model Organisms to Study Genetics
- Quiz & Worksheet - Graph the Derivative from Any Function
- Quiz & Worksheet - Difference Between Asexual and Sexual Reproduction

- Photolysis and the Light Reactions: Definitions, Steps, Reactants & Products
- Schizophrenic Delusions: Definition, Types & Symptoms
- Haitian Revolution Lesson Plan
- Caps for Sale Lesson Plan
- What is the CSET Exam?
- How to Assign a Study.com Lesson to Your Students
- What are Passing Scores for GACE Tests?
- Math Card Games for Kids
- How Much Does the NCLEX Cost?
- Concept Attainment Lesson Plan
- Political Spectrum Lesson Plan
- Apartheid Lesson Plan

Browse by subject