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

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 79 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

- DSST Principles of Advanced English Composition: Study Guide & Test Prep
- Upper Level SSAT: Test Prep & Practice
- The Adventures of Sherlock Holmes Study Guide
- Sherlock Holmes Short Stories Study Guide
- PTE Academic Test: Practice & Study Guide
- Finding, Evaluating & Using Sources
- Revising & Editing an Essay
- Citing & Documenting Sources
- Analyzing Arguments in Writing
- Audience & Goal In Writing
- TOEIC Listening & Reading Test: Purpose & Format
- Excelsior College BS in Business Degree Plan Using Study.com
- IELTS General Training Reading: Format & Task Types
- IELTS General Training Writing: Format & Task Types
- Gates-MacGinitie Reading Test Scores
- IELTS General Training Test: Structure & Scoring
- Supply and Demand Activities for Kids

- The Sherman Antitrust Act of 1890: Summary & Overview
- Cost of Living Adjustment: History & Formula
- Trends in Social Media Marketing
- Interest Rate Risk: Definition, Formula & Models
- Special Education Transition Plans from Middle School to High School
- Spartan Traditions: Festivals & History
- American Colonial Music: Instruments & Facts
- Strategies for Teaching Manners to Students with Autism
- Quiz & Worksheet - The Holistic View in Anthropology
- Quiz & Worksheet - Changes & Updates to Project Schedules
- Quiz & Worksheet - Hotel Housekeeping Basics
- Quiz & Worksheet - Capital Requirements
- Quiz & Worksheet - Superposition Theorem
- Muscle Contraction Flashcards
- Water Polo Flashcards

- American Revolution Study Guide
- Financial Accounting: Homework Help Resource
- SAT Subject Test Mathematics Level 2: Practice and Study Guide
- Introduction to Nutrition: Certificate Program
- College Algebra Remediation
- Chapter 5: Projectile Motion
- Universe Theories Lesson Plans
- Quiz & Worksheet - Math Equations
- Quiz & Worksheet - Impact of Emotion on Behavior
- Quiz & Worksheet - When to Use Group or Individual Decision Making
- Quiz & Worksheet - Characteristics of Complementary Goods
- Quiz & Worksheet - Calculating Producer Price Index

- What is Deflation? - Definition, Causes & Effects
- What is Segregation? - Definition, Facts & Timeline
- Special Education Laws in Florida
- 8th Grade Persuasive Writing Prompts
- CFA (Chartered Financial Analyst) Exam Dates & Registration
- Disability Advocacy Groups
- Best GMAT Prep Book
- Point of View Lesson Plan
- How to Save for College
- Black History Month for Kids
- How to Get an AP Exam Fee Waiver
- Broward County Adult Education

Browse by subject