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Rational Expressions Calculator

Select the operation and provide the rational expression with or without operation available. The calculator will instantly apply arithmetic operations on it and simplify to the most reduced form.

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An online rational expressions calculator factorizes the given function and performs various mathematical operations to reduce it to a most simplified form. You can perform more than one arithmetic actions at the same time on a given problem if applicable. This content is arranged with proper organic information that is very crucial in optimizing a given rational expression. Look over it!

What Is A Rational Expression?

In the dictionary of algebraic expressions:

“A fraction containing numerator and/or denominator in the form of algebraic polynomials is called a rational expression”

There is nothing complex to understand as it is a generic form of fraction that consists of simple or complex rational functions. A better use of a free online rational function calculator helps you to solve such problems in a glimpse of an eye.

For example:

$$ \frac{6x - 1}{{9x^{2} - 1}}\hspace{0.25in}\,\,\,\,\,\frac{{{x^5} + 6x + 1}}{{{x^2} - 7x - 2}} $$


You can consider a polynomial as a rational expression too. Suppose we have the following polynomial: $$ 2 x^{2} + 7x + 50 $$ What if we write this as follows: $$ \frac{{2{x^2} + 7x + 50}}{1} $$ It seems a little bit odd but can be considered as a rational expression.

This is the reason why you can consider a polynomial as a standard rational function. You can reduce a complex polynomial by using a free online rational number calculator. What you need to do is to suppose 1 in such cases for sure.

Operations On Rational Expressions:

Here, we have a series of algebraic operations need to be performed on rational expressions. These include:


You can add two or more rational expressions with the help of a free adding rational function calculator. But when it comes to manual calculations, you have to figure out common factors and cancel them to get the reduced form. What you need to do is here as follows:

  • Write all separate terms as a sum.
  • Take a common denominator for all expressions by subjecting to least common multiple(LCM).
  • Now, add all the terms in the numerator with the denominator remain the same.
  • If you find any similar terms with opposite signs, cancel them and rewrite all rest terms in a proper sequence.
  • You get the simplified form.


Subtracting two or more rational polynomials is exactly opposite to that of addition as it is defined for numbers. A free subtracting rational expression calculator may assist you to perform subtraction of two or more rational functions. Also, you should follow these rules to subtract rational functions.

  • Write all separate terms as a subtraction.
  • Take a common denominator for all expressions with the help of LCM.
  • Subtract all the terms in the numerator.
  • Cancel all those having opposite signs with the same variables and add the rest with the same sign and variable powers.
  • Make it clear that the denominator remains unaltered.
  • This is how simplified form is obtained.


You can multiply rational polynomials just like you do it for numbers. But do not forget to follow guidelines below:

  • Write all expressions with a multiplication sign among them.
  • Take a product of all values in the numerator and denominator separately.
  • Now, multiply further each factor in the values following the basic distributive property
  • Add all terms with same signs and variables and subtract those with opposite signs.
  • Rewrite the expression in an descending order of variable power.
  • Simplified rational expression is obtained.


Just like addition and subtraction relationship, multiplication and division of two or more rational expressions is also the same. When you come across some difficult terms, you can use our free online dividing rational expressions calculator to reduce them instantly. You will definitely get step-by-step calculations to avoid any inconvenience. But in practice, you must pay attention to the following points.

  • Write all terms with a division sign in between.
  • Except the first term, replace numerator and denominators of all terms and change division sign to multiplication sign.
  • For rest calculations, follow the same rules as defined for multiplication of rational polynomials.

How To Simplify Rational Expressions?

You can use our free online simplifying rational function calculator to reduce the complex terms involved in the expressions to a reduced one. But when it needs to be simplified manually, we have defined all rules and regulations for that above. For instance, let us solve a few examples corresponding to each of the above operations.

Example # 01: Reduce the terms in the rational expression given below: $$ \frac{x^{2} - 6 x + 9}{\left(x + 1\right) \left(x^{2} - 1\right)} $$


The given rational function is:

$$ \frac{x^{2} - 6 x + 9}{\left(x + 1\right) \left(x^{2} - 1\right)} $$

Remember the following formulae:

$$ \left(a - b\right)^{2} = a^{2} -2ab + b^{2} $$ and $$ \left(a^{2} - b\right) b = \left(a + b\right)(\left(a - b\right) $$

We apply these formulae on the above function to reduce it.

$$ =\frac{ \left(x - 3\right)^{2}}{\left(x + 1\right)\left(x + 1\right)\left(x - 1\right)} $$

$$ =\dfrac{\left(x - 3\right)^{2}}{\left(x - 1\right) \left(x + 1\right)^{2}} $$

Which is the most simplified form of the given function. Here, our free simplify rational expressions calculator determines the reduced form of any rational polynomial quickly.

How Rational Expression Calculator Works?

You may adopt many methods to simplify a rational expression but a free rational numbers calculator is the best among all. No matter how complicated the given function is, our calculator optimizes it into a most generic form. Let us guide you how to do it!

Input: First, select either of the following options from the menu list:

(1) Reduced terms of rational expression

(2) Adding, subtracting, multiplying and dividing of rational expressions

(3) Simplify any expressions

If you select option 1:

  • Enter the numerator function
  • Enter the denominator function
  • Click ‘calculate’

If you choose option 2:

  • First select either you need to apply operations on two or three expressions
  • After doing so, write down numerator and denominator expressions against each rational function in the input field
  • Hit the calculate button.

If you choose option 3:

  • Write down your input function in the menu bar
  • Click 'calculate’ button

Output: The free rational expression calculator performs the following operations according to the input selected for the rational expressions.

For option 1:

  • Reduces the rational expression step by step for a generic answer

For option 2:

  • Adds, subtracts, multiplies and divides two or three functions, respectively.

For option 3:

  • Simplifies the whole rational expression and results in the most optimized form of the rational expression given.


How do you know if the rational expression is proper or improper?

Proper rational expression: “A rational expression in which numerator has highest degree of the variable instead of the variable in the denominator”

Improper rational expression: “A rational function in which the degree of the numerator is less than that of the degree of the variable in the denominator is said to be improper.” No matter what type of rational expression it is, use a free online rational function calculator to simplify it in a fraction of seconds.

What is meant by the degree of the polynomial?

The highest degree of a variable involved in a polynomial is said to be the degree of the polynomial.

For example: Consider the following polynomial below:

$$ 6x^{9} + 4x^{5} + 5x^{4} + 5x^{2} + 10 $$

As we see that for individual terms in the above polynomial, the degrees are 9, 5, 4 and 2, respectively. But the highest degree is 9. That is why it will be considered as the degree of the whole polynomial. Also, an online rational expressions calculator helps you to further simplify this polynomial function properly.

What is meant by monomial?

An algebraic sentence that contains only one term is known as a monomial. For example: 5z, 4x, 65y etc.


Simplifying rational expressions makes it possible to reduce complexity involved in your problems. Mathematicians, Data scientists, Engineers and Physicists make a vast use of rational expressions where they use a free online rational functions calculator to perform quick calculations.


From the source of wikipedia: Algebraic fraction, Rational fractions, Irrational fractions, Terminology. From the source of khan academy: Reducing rational expressions, End behavior, Discontinuities of rational functions. From the source of lumen learning: The Domain of a Rational Function, Asymptotes, Simplifying a Rational Expression, Partial Fractions.

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