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 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

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
3 in chapter 8 of the course:

Back To Course

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

- Introduction to HTML & CSS
- Introduction to JavaScript
- Computer Science 332: Cybersecurity Policies and Management
- Introduction to SQL
- Computer Science 203: Defensive Security
- JavaScript Language Basics
- Error Handling, Debugging & Events in JavaScript
- HTML Elements & Lists
- Conditionals, Arrays & Loops in JavaScript
- Introduction to HTML
- Anti-Bullying Survey Finds Teachers Lack the Support They Need
- What is the ASCP Exam?
- ASCPI vs ASCP
- MEGA Exam Registration Information
- MEGA & MoGEA Prep Product Comparison
- PERT Prep Product Comparison
- MTLE Prep Product Comparison

- Simple Scientific Tools & Uses for Kids
- Chi Square Distribution: Definition & Examples
- Stars: Definition & Facts
- Linear Approximations Using Differentials: Definition & Examples
- Access Control: Types & Implementation
- 'I Am' Poem Lesson Plan
- Key Controls in Cybersecurity Risk Management: Definition & Use
- Quiz & Worksheet - Line Integrals
- Quiz & Worksheet - Frankenstein Creature Quotes
- Quiz & Worksheet - A Christmas Carol Facts
- Quiz & Worksheet - Preschool Classroom Technology
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies
- Music Lesson Plans
- Calculus Worksheets

- Neuroanatomy: Help & Tutorials
- UExcel Human Resource Management: Study Guide & Test Prep
- PLACE Marketing Education: Practice & Study Guide
- Western Civilization from 1648 for Teachers: Professional Development
- NYSTCE Communication & Quantitative Skills Test (080): Practice & Study Guide
- Setting Goals & Objectives
- CEOE Middle Level/Intermediate Math: Sequences & Series
- Quiz & Worksheet - Density of Liquids Facts for Kids
- Quiz & Worksheet - Anti-anxiety Drugs & Sedative Hypnotics
- Quiz & Worksheet - Organizational Communication at Dell
- Quiz & Worksheet - Camazotz in A Wrinkle in Time
- Quiz & Worksheet - Diffraction & Huygen's Principle

- Grasshopper Facts: Lesson for Kids
- Benefits of Exercise for Children
- Science Project Grading Rubrics
- What is the LSAT?
- Maximizing Performance on SAT Subject Tests
- Science Projects for Preschoolers
- Body Language During a Job Interview
- Chinese New Year Lesson Plan
- How Much Does it Cost to Study Abroad?
- 4th Grade Math Centers
- Preparing for the AP Biology Exam: Tips & Tricks
- States that Require Physical Education

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

Browse by subject