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 70,000 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 brings everything you learned about fractions into the world of algebra. We will mix common denominators with factoring and FOILing.

The word 'rational' means 'fraction.' So a rational polynomial is a fraction with polynomials in the numerator (top) and/or denominator (bottom). Here's an example of a rational polynomial:

(*x* + 4) / (*x*^2 + 3*x* + 2)

As we get started, let's 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 get started!

- We need to factor.
- Find a common denominator.
- Rewrite each fraction 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.

There are quite a few steps, but let me show you how they work.

Our first expression is (1 / (*x* - 2)) + (3 / (*x* + 4)).

The first step is to factor. Since we don't have anything to factor, let's move to the next step, writing down our denominators, (*x*-2) and (*x*+4). This will be our common denominator: (*x* - 2)(*x* + 4).

Now we need to create our common denominator. Let's look at our first term, (1 / (*x* - 2)). (*x* - 2) is in the denominator. We need to multiply by (*x* + 4) to make our common denominator. But if we multiply by (*x* + 4) on the bottom, we need to multiply by (*x* + 4) on the top.

For right now we are going to write it and not multiply yet.

Let's look at our second term: (3 / (*x* + 4)). The denominator is (*x* + 4). We need to multiply (*x* - 2) times (*x* + 4) to get our common denominator. But once again, if we multiply by (*x* - 2) on the bottom, we need to multiply by it on the top too.

So far, this is what we have:

((1(*x* + 4)) / ((*x* - 2)(*x* + 4))) + ((3(*x* - 2)) / ((*x* + 4)(*x* - 2)))

Don't FOIL the denominator - we may have to cancel as our final answer!

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

(1(*x* + 4)) + 3(*x* - 2)) / ((*x* - 2)(*x* + 4))

Let's simplify the numerator.

1(*x* + 4) = *x* + 4

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

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

Collect like terms in the numerator.

(4*x* - 2) / ((*x* + 4)(*x* - 2))

Factor the numerator if possible.

4*x* - 2 = 2 (2*x* - 1)

(2(2*x* - 1)) / ((*x* + 4)(*x* - 2))

There isn't anything to slash or cancel, so we distribute and FOIL for our final answer.

(4*x* - 2) / (*x*^2 + 2*x* - 8)

((2*x*) / (*x*^2 - 16)) - (1 / (*x* + 4))

*x*^2 - 16 factors into (*x* - 4)(*x* + 4). So let's put that into the expression.

((2*x*) / ((*x* - 4)(*x* + 4))) - (1 /(*x* + 4))

Our next step is to write down all of our denominators.

In the first term, we have (*x* + 4)(*x* - 4), so we write those down.

We continue to the next term and look at the denominator. We never duplicate denominators from term to term. Since we already have (*x* + 4) written as part of our denominator, we don't need to duplicate it. So it turns out our common denominator will be (*x* + 4)(*x* - 4).

Now we need to create our common denominator. Let's look at our first term ((2*x*) / (*x* + 4)(*x* - 4)). We already have our common denominator here, so we're going to move to the next term: (1 / (*x* + 4)).

Here, we need to multiply (*x* - 4) to make our common denominator. But if we multiply (*x* - 4) on the bottom, we need to multiply by (*x* - 4) on the top. For right now, we are going to write it and not multiply yet. So we have ((2*x*) / (*x* + 4)(*x* - 4)) - (1(*x* - 4) / (*x* + 4)(*x* - 4)).

Let's write the numerator all over the denominator.

((2*x*)-1(*x*-4))/((*x*+ 4)(*x* - 4))

Simplify the numerator (or top) and rewrite it over the denominator.

Distribute the -1 into (*x* - 4) = -1*x* + 4.

Collecting like terms, 2*x* - 1*x*= *x*.

So now our expression looks like:

(*x* + 4) / (*x*+ 4)(*x*- 4)

We can slash, or cancel, (*x*+ 4) over (*x*+ 4).

This gives us 1/(*x* - 4) as our final answer.

((5*x*^2 - 3) / (*x*^2 + 6*x* + 8)) - 4

The first step is to factor.

*x*^2 + 6*x* + 8 = (*x* + 4)(*x* + 2)

Our next step is to write down all of our denominators.

In our first term, we have (*x* + 4)(*x* + 2), so we write it down.

The denominator for the next term is 1.

Therefore, our common denominator will be (*x* + 4)(*x* + 2).

Now we need to create our common denominator. Let's look at our first term (5*x*^2 - 3)/((*x* + 4)(*x* + 2)).

We already have our common denominator here, so we're going to move to the next term, 4.

Here, we only have a 1 in the denominator, so we need to multiply by (*x* + 4)(*x* + 2) over (*x*+4)(*x*+2).

This is what our new expression is going to look like:

((5*x*^2 - 3) / (*x* + 4)(*x* + 2)) - ((4 (*x* + 4)(*x* + 2)) / ((*x* + 4)(*x* + 2))) .

Let's write the whole numerator (top) over the denominator (bottom).

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

We can now simplify the top, or numerator.

(*x*+4)(*x*+2) = *x*^2 +6*x* +8

Multiply -4( *x*^2 +6*x* +8) and we have -4*x*^2 - 24*x* - 32.

Let's continue with the numerator and collect like terms, so our expression looks like:

(*x*^2 - 24*x* - 35) / ((*x* + 4)(*x* + 2))

The numerator does not factor without using the quadratic formula, so this is almost our answer, except we need to FOIL the bottom, or denominator. Here is our final answer:

(*x*^2 - 24*x* - 35) / (*x*^2 + 6*x* + 8)

As we have seen, the process to add or subtract rational expressions is:

- We need to factor.
- Find a common denominator.
- Rewrite each fraction 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.

Once you complete this lesson you'll be able to add or 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
3 in chapter 8 of the course:

Back To Course

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

- Inclusion in Recruitment, Interviews & Hiring
- Computer Science 105: Introduction to Operating Systems
- High School 101: High School Readiness
- Communications 301: Diversity and Intercultural Communication
- Communications 106: Communication in the Digital Age
- Operating System Fundamentals
- Cultural Differences in Nonverbal Communication
- Techniques for Inclusive Hiring & Onboarding
- Intro to Inclusion in Recruitment, Interviews & Hiring
- Implementing Inclusion in Recruitment & Screening
- CLEP Prep Product Comparison
- CLEP Exam vs. AP Test: Difficulty & Differences
- CLEP Tests for the Military
- How to Transfer CLEP Credits
- CLEP Exam Question Formats
- CLEP Exam Costs & Registration Deadlines
- CLEP Exam List & Credits Offered

- Static vs. Non-Static Methods in Java
- South African Flag Lesson for Kids: Colors & Meaning
- Monkey Facts: Lesson for Kids
- Peter Senge: Learning Organizations & Systems Thinking
- Students with Dual-Sensory Impairment: Characteristics & Accommodations
- Pharmaceutical Compounding: Equipment & Supplies
- How Physiology of the Brain Affects Emotional Intelligence
- Risk Reporting: Communication & Techniques
- Quiz & Worksheet - The Crucible & Symbols
- Quiz & Worksheet - Various Perspectives on Organizations
- Animal Adaptations: Quiz & Worksheet for Kids
- Quiz & Worksheet - Serving Alcohol Responsibly
- Quiz & Worksheet - Email Message Anatomy
- How to Cite Sources Flashcards
- Evaluating Sources for Research Flashcards

- Common Core Math - Geometry: High School Standards
- ILTS Social Science - History: Test Practice and Study Guide
- Math 105: Precalculus Algebra
- Physical Geology for Teachers: Professional Development
- French Revolution Study Guide
- NMTA Social Science: The U.S. Court System
- PLACE Business Education: Marketing Ethics and Law
- Quiz & Worksheet - Impacts of Behavioral Sciences on Organizational Behavior
- Quiz & Worksheet - Types of Cultural Perceptions of Time in Organizations
- Quiz & Worksheet - Figurative Language in Poetry
- Quiz & Worksheet - Master Slides & Layouts in PowerPoint
- Quiz & Worksheet - Multiple Choice Questions on the AP Chemistry Exam

- How to Add and Modify Images in PowerPoint: Resizing, Cropping and Style
- Trace Fossil: Definition, Examples & Importance
- Telling Time Games & Activities
- How to Get AP Exam Credit
- How Many Times Can You Take the TEAS Test?
- What Is the PSAT 10? - Information, Structure & Scoring
- How to Ace Your Job Interview
- Best Psychology Books for Undergraduates
- English Language Learning Programs in California
- What Is History? - Lesson Plan
- How to Flip Your Classroom with Study.com
- WIDA Can Do Descriptors for Kindergarten

Browse by subject