Rational Expressions Simplification – A Brief Guide

Mathematics, a gem in itself, has numerous dimensions today for simplifying various problems. No doubt some of them are easy to understand, but still, a large community of pupils around the globe still hates mathematics just because of the complexity involved in the subject. Just take an example of algebraic expressions. What if your professor asks you to simplify an expression written either simply or division or any other form. You are not having any idea about it. What to do? Do not worry at all as the free rational expression calculator by calculator has made it possible to get a firm grip over such complicated problems. 

Gone are the days when simplifying rational expression was considered one of the most complex tasks. Now it could be done immediately with the assistance of the free online rational equation calculator. How does it sound to you? Really great? Yes, it definitely is.

Okay, well enough! Let us come to the topic now. In this context below, we will be taking you through the encyclopaedia of rational expressions. Moreover, we will highlight ways and rules to simplify them and use a rational or irrational calculator to speed up the calculations. So anxious to know more?  Let’s go!

What Is a Rational Expression?

In the context of algebraic analysis:

A rational expression is considered a fraction of two polynomials. 

Here if f is a rational expression, then you would rather say that it is the ratio of a couple of polynomials say p and q. 

F = p/q

It is a general form of the fraction that includes simple or complicated rational functions, thus it is not difficult to grasp. A greater understanding of how to utilise a free online rational or irrational calculator will help you answer such difficulties in a flash. 

The basic arithmetic operations, including addition, subtraction, multiplication, and division, may be performed on rational expressions just like they can on polynomials or any other sort of expression. When any of these operations are done on two rational expressions, the outcome is always another rational expression, which is a pleasant characteristic of rational expressions. In contrast to polynomials, multiplying and dividing two rational expressions is often straightforward, while adding and subtracting two rational expressions is challenging.


The following given expressions are rational expressions indeed and can be easily simplified with the usage of a free online rational or irrational calculator

x – 2x + 1,    x3 – 2x2 + x + 1,   y2 – y + 5

Look at the last expression. What do you notice? Do you feel that it is not a rational expression due to no denominator? Well, let us tell you that if there is no polynomial in the denominator, it will be considered 1 always. That is why this expression is also considered a rational one. Moreover, you can also make use of the free online rational expression calculator to simplify the rational expressions quickly and precisely. No matter how complicated the expressions are, this free calculator will get it done for you and display the final results on your device screen in seconds. Isn’t it a great offer? Yes, it is!  

Procedures on Rational Expressions: 

Here, a sequence of algebraic operations on rational expressions must be done.

These are some of them:


Using a free rational expression calculator, you may combine two or more rational expressions. However, while performing manual calculations, you must identify common elements and cancel them to obtain the reduced version.

The following is what you must do:

  • Add together all of the distinct terms to get a total.
  • Apply the least common multiple to all phrases to find a common denominator (LCM).
  • Add all of the terms in the numerator while keeping the denominator the same.
  • If you come across any comparable phrases with opposing signs, delete them and rewrite the rest of the terms in the correct order.
  • This is how you will get the simplified form of rational expression.


Subtracting two or more rational equations is the inverse of addition in the sense that it is defined for integers. Subtracting two or more rational functions may be made easier with the help of a free rational or irrational calculator. To subtract rational functions, you should also follow these requirements.

  • Subtract all of the terms separately.
  • Using LCM, find a common denominator for all expressions.
  • Subtract all of the numerator terms.
  • All those with opposing signs and the same variables should be cancelled, and the remainder should be added with the same sign and variable powers.
  • Ensure that the denominator remains unchanged.
  • This is how you get a simpler form.


Multiplying rational polynomials is similar to multiplying integers. But don’t forget to adhere to the following guidelines:

  • Include a multiplication sign in all of your expressions.
  • Take the product of every numerator and denominator value separately.
  • Now, using the fundamental distributive property, multiply each component in the values again.
  • Subtract all terms and variables with the same sign and add those with the opposite sign.
  • Rewrite the formula in the order of variable power in decreasing order.
  • This is how it becomes possible to derive a simplified rational expression.


Multiplication and division of two or more rational expressions have the same connection as addition and subtraction. When you come across some challenging phrases, you may quickly decrease them using a free online rational expressions calculator. To avoid any hassle, you will undoubtedly receive step-by-step computations. However, you must pay attention to the following issues in practice.

  • All words should be separated by a division symbol.
  • Replace the numerator and denominators in all terms except the first, and convert the division sign to a multiplication sign.
  • Apply the same rules for rest calculations as you did for rational polynomial multiplication.

Types of Rational Expressions:

Basically, there are a couple of types of the rational expressions that are as follows:

Proper Rational Expression:

A particular expression in which the degree of the variable in the numerator becomes higher than the degree of variable in the denominator is called a proper rational expression.


x2 + x – 3x – 5

As you see in the above expression, the power of the variable x is high in the polynomila above line bar. This is why the ratio is known as the proper rational expression.

Improper Rational Expression:

If the power of the variable in the denominator exceeds that of the numerator, then we say that the given expression is the improper rational expression.


x2 + x – 3x3 – 5

How To Simplify Rational Expressions:

Here we will resolve basic to complex rational expression problems that will let you understand the concept clearly and accurately. Rest of the speed can be adopted by using an online rational or irrational calculator. Anyways, let’s pay heed to what is going to happen next!

Problem # 01:

Simplify the following rational expression:

x2 – 9x – 3


The given rational expression is improper one. So this is how it will get reduced:

x2 – 9x – 3

= (x + 3)(x – 3)(x – 3)

= (x + 3) as the expression (x – 3) will be cancelled out. 

Problem # 02:

Resolve the rational expression ratio given below:

x2 + 2x – 3x2 – 3x – 3 4x + 2x2 – x 2x2 – 9x – 5x2 – 2x – 1


The given expression can be simplified as follows:

x2 + 2x – 3x2 – 3x – 3 4x + 2x2 – x 2x2 – 9x – 5x2 – 2x – 1 

= x2 + 2x – 3x2 – 3x – 3 x2 – x4x + 2 2x2 – 9x – 5x2 – 2x – 1

= (x + 3)(x – 1)(x – 5)(x + 2) x(x – 1)2(2x + 1) (2x + 1)(x – 5)(x – 1)2

On canceling all identical terms, we are only left with the following ones:

= x(x + 3)2(x + 2)

Which is the required reduced form. In order to keep your computations faster, you can be subject to a free online rational or irrational calculator. 

Problem # 03:

Add a couple of rational expressions given below:

xx2 – 4 + 2x2 – 4


Here we have:

xx2 – 4 + 2x2 – 4

= x + 2x2 – 4

= x + 2(x + 2)(x – 2)

= 1x – 2

Which is the required answer and could also be verified by using a free online rational expression calculator. 

Problem # 03:

A swimming pool’s input pipe may be utilised to fill the pool in 12 hours. In 20 hours, the drain pipe may be used to empty the pool. How long will it take to fill the pool if it is empty and the drain line is unintentionally opened?


From the statement given above, we have the following data:

Inlet pipe = 12 hours

Drain pipe = 20 hours

Together = x hours

1/12 – 1/20 = 1/x

60x(1/12 – 1/20) = 60x(1/x)

5x – 3x = 60

2x = 60

x = 60/2

x = 30

The Domain of a Rational Expression:

We presume that the domain of a function is the set of all real numbers for which the function is defined unless otherwise stated. When the denominator is 0, a rational expression is undefined. As a result, determining all values (if any) that cause the denominator to be zero is required for determining the domain of a rational function.

Not only this, but the free rational or irrational calculator also goes for determining the domain and range of any expression in a couple of seconds and without any errors. 

Use of Rational Expression Calculator:

The free and fast result oriented rational calculator by calculator has made it far possible to resolve various expressions in no time and efficiently. Let’s find how!

First, choose one of the following options from the drop-down menu:

(1) Reduced rational expression words

(2) Rational expressions: adding, subtracting, multiplying, and dividing

(3) Make any phrases as simple as possible.

If you choose option 1, you will be presented with the following options:

  • Add the numerator function to the equation.
  • Add the denominator function to the equation.
  • Select ‘compute’ from the drop-down menu.

If you select option 2, you will be presented with the following options:

  • First, decide whether you’ll be working with two or three expressions.
  • Then, against each rational function in the input box, write down numerator and denominator expressions.
  • To calculate, use the calculate button.

If you select option 3, you will be presented with the following options:

  • In the menu bar, jot down your input function.
  • To compute, use the ‘calculate’ button.

According to the rational expressions provided, the free rational or irrational calculator performs the following procedures.

For the first option:

For a general solution, reduce the rational statement step by step.

Option number two:

Two or three functions are added, subtracted, multiplied, and divided.

Option three:

Simplifies the entire rational expression, yielding the most efficient form of the provided rational expression.

How fast and accurate is that!

Fractions of a Complex:

A simple fraction is any rational expression with no rational expression in the numerator or denominator.

A complex fraction is any rational expression with a rational expression in the numerator or denominator.

To make difficult fractions easier to understand, use the following formula:

Step 1: 

Locate the LCD and identify all fractions in the numerator and denominator.

Step 2: 

Multiply the LCD by the numerator and denominator.

Wrapping It Up:

Rational equations can be utilised to answer a wide range of rate, time, and work-related problems. We may use rational expressions and equations to answer concerns about how to combine employees or robots to accomplish a project on time by using rational expressions and equations. 

In this guidepost, we had a discussion regarding rational expression simplification along with the use of free rational expression calculator by calculator online. So stop thinking and start using this free calculator for instant outputs straightaway. We hope his post is going to be something of benefit for you. 

Good Luck!

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