Back To Course

Math 101: College Algebra12 chapters | 95 lessons | 11 flashcard sets

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

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

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 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 | 95 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

- Computer Science 109: Introduction to Programming
- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Early Civilizations & The Ancient Near East
- Fundamental Overview of World War I
- The Virginia Dynasty & Jacksonian America
- 1920's America and the Great Depression
- Building the United States After the American Revolution
- CEOE Test Cost
- PHR Exam Registration Information
- Claiming a Tax Deduction for Your Study.com Teacher Edition
- What is the PHR Exam?
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP

- Subtraction in Java: Method, Code & Examples
- Hydrogen Chloride vs. Hydrochloric Acid
- Extraction of Aluminum, Copper, Zinc & Iron
- Iroquois Culture, Traditions & Facts
- Noun Clauses Lesson Plan
- Adverb of Manner Lesson Plan
- Timeline Project Ideas for High School
- Quiz & Worksheet - Multi-Dimensional Arrays in C
- Quiz & Worksheet - What is a Diastereoisomer?
- Quiz & Worksheet - Mauryan Empire Art & Culture
- Quiz & Worksheet - What is a Convergent Sequence?
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- 2nd Grade Math Worksheets & Printables
- Cyberbullying Facts & Resources for Teachers

- How to Get an Internship
- AP English Language: Homeschool Curriculum
- How to Get an Internship
- NMTA Essential Academic Skills Subtest Math (003): Practice & Study Guide
- High School US History: Homework Help Resource
- OAE Reading: Informational Texts
- Inflation & Unemployment - MTEL Political Science/Political Philosophy
- Quiz & Worksheet - Hinduism Characteristics & History
- Quiz & Worksheet - What Is First Degree Murder?
- Quiz & Worksheet - The Bronze Age
- Quiz & Worksheet - African Religious Systems
- Quiz & Worksheet - The Sasanid Empire

- What is a Static Character? - Definition & Examples
- Marie Curie Discoveries
- Jobs for Teachers Outside of Education
- Average GMAT Scores & Percentiles
- How to Learn English Online
- How to Ace a Second Interview
- Persuasive Writing Prompts: Middle School
- Top 5 Education Documentaries
- Math Fraction Games
- Mental Math Games
- Learning Activities for Children with Down Syndrome
- Preschool Book List

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject