•

Dec 1, 2025

•

4 pages

Understanding Rational Expressions and Equations

Layla Guthrie

@laylaguthrie_aoly

Rational expressions are fractions where both the numerator and denominator... Show more

1 / 4

RATIONAL EXPRESSIONS
& EQUATIONS
・A rational expression is the ratio of two polynomials
р ахо
Evaluate
x
X-3
(-3)-3---3-1-1/21
Finding the R

Understanding Rational Expressions

A rational expression is simply a fraction where both parts are polynomials, written as $\frac{p}{q}$ where $q ≠ 0$ . When evaluating these expressions, just substitute the variable value and calculate.

Before working with rational expressions, you need to identify restricted values—values that make the denominator zero (which would cause division by zero). To find these, set the denominator equal to zero and solve. For example, in $\frac{y-3}{2y+7}$ , setting $2y+7=0$ gives us $y = -\frac{7}{2}$ as the restricted value.

When simplifying rational expressions, factor both numerator and denominator completely, then cancel common factors. Like in $\frac{2p-14}{p^2-49}$ , we can rewrite this as $\frac{2(p-7)}{(p+7)(p-7)}$ , which simplifies to $\frac{2}{p+7}$ since the $(p-7)$ terms cancel.

Pro Tip: Always identify restricted values before simplifying! This prevents losing critical information about where the expression is undefined.

Simplifying and Multiplying Rational Expressions

Simplifying rational expressions often requires factoring skills. For instance, $\frac{5-y}{y^2-25}$ can be rewritten as $\frac{-(y-5)}{(y+5)(y-5)}$ , which simplifies to $\frac{-1}{y+5}$ . Notice how recognizing the difference of squares pattern in the denominator was key!

When multiplying rational expressions, follow these steps:

Factor all numerators and denominators completely
Cancel any common factors between numerators and denominators
Multiply the remaining factors in the numerators and denominators separately

For example, when multiplying $\frac{3c-3d}{6c} \cdot \frac{2}{c^2-d^2}$ , first rewrite as $\frac{3(c-d)}{6c} \cdot \frac{2}{(c-d)(c+d)}$ . The common factor $(c-d)$ cancels, leaving $\frac{1}{c} \cdot \frac{1}{c+d} = \frac{1}{c(c+d)}$ .

Remember: The formula for multiplying rational expressions is $\frac{p}{q} \cdot \frac{r}{s} = \frac{pr}{qs}$ – but only after you've factored and simplified!

Division of Rational Expressions

Division with rational expressions isn't as scary as it seems! The key trick is to convert division into multiplication by using the reciprocal of the divisor.

To divide rational expressions, flip (take the reciprocal of) the second fraction and change the division to multiplication. For example, $\frac{5+15}{2} \div \frac{+2-9}{10}$ becomes $\frac{5+15}{2} \cdot \frac{10}{+2-9}$ .

After converting to multiplication, follow the same process: factor completely, cancel common factors, then multiply remaining terms. This works for complex expressions too!

When working with multiple divisions like $\frac{3x}{4y} \div \frac{3x}{4y} \div \frac{5x}{6y}$ , handle them one at a time, working from left to right. Converting each division to multiplication makes these problems much more manageable.

Quick Tip: Division is just multiplication by the reciprocal! Always flip the second fraction and change ÷ to ×.

Addition and Subtraction of Rational Expressions

Adding or subtracting rational expressions is easiest when the denominators are the same. With identical denominators, you can simply combine the numerators while keeping the denominator: $\frac{p}{q} + \frac{r}{q} = \frac{p+r}{q}$ .

For example, $\frac{2}{3d+5} + \frac{7d}{3d+5} = \frac{2+7d}{3d+5}$ or $\frac{7d+2}{3d+5}$ . Notice that you don't cancel anything here—the numerator and denominator don't share common factors.

When denominators differ, you'll need to find the least common denominator (LCD). The LCD contains all factors from all denominators, each raised to its highest occurring power. For fractions like $\frac{5}{14}$ , $\frac{3}{49}$ , and $\frac{1}{8}$ , factor the denominators first: $\frac{5}{2 \cdot 7}$ , $\frac{3}{7^2}$ , $\frac{1}{2^3}$ . The LCD would be $7^2 \cdot 2^3 = 392$ .

Heads Up: When adding or subtracting rational expressions, don't try to cancel terms in the numerator with terms in the denominator! This only works with multiplication and division.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Algebra 2

•

168

•

Dec 1, 2025

•

4 pages

Understanding Rational Expressions and Equations

Layla Guthrie

@laylaguthrie_aoly

Rational expressions are fractions where both the numerator and denominator are polynomials. Understanding how to work with these expressions—evaluating, simplifying, and performing operations with them—is an essential algebra skill that builds your mathematical toolkit for more advanced topics.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Understanding Rational Expressions

Pro Tip: Always identify restricted values before simplifying! This prevents losing critical information about where the expression is undefined.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Simplifying and Multiplying Rational Expressions

When multiplying rational expressions, follow these steps:

Factor all numerators and denominators completely
Cancel any common factors between numerators and denominators
Multiply the remaining factors in the numerators and denominators separately

Remember: The formula for multiplying rational expressions is $\frac{p}{q} \cdot \frac{r}{s} = \frac{pr}{qs}$ – but only after you've factored and simplified!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Division of Rational Expressions

Division with rational expressions isn't as scary as it seems! The key trick is to convert division into multiplication by using the reciprocal of the divisor.

After converting to multiplication, follow the same process: factor completely, cancel common factors, then multiply remaining terms. This works for complex expressions too!

Quick Tip: Division is just multiplication by the reciprocal! Always flip the second fraction and change ÷ to ×.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Addition and Subtraction of Rational Expressions

For example, $\frac{2}{3d+5} + \frac{7d}{3d+5} = \frac{2+7d}{3d+5}$ or $\frac{7d+2}{3d+5}$ . Notice that you don't cancel anything here—the numerator and denominator don't share common factors.

Heads Up: When adding or subtracting rational expressions, don't try to cancel terms in the numerator with terms in the denominator! This only works with multiplication and division.