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Rational Expression Calculator + Online Solver With Free Steps

The Rational Expression Calculator is an online tool that is very handy and is used to simplify given rational expressions and functions.

Solving and simplifying the complex rational expression is a tedious and time-consuming task. Still, with our free online Rational expression calculator, you can solve complex rational expressions quickly and easily.

The result is displayed in the form of a simplified fraction. The calculator also gives the option of viewing detailed solutions with steps to have a better understanding.

What Is a Rational Expression Calculator?

A Rational Expression Calculator is an online calculator that can be used to solve any kind of rational expressions in just a few seconds.  

The Rational Expression calculator displays the simplified and rationalized form of any given fraction containing polynomials.

It uses the factorization technique to rationalize the given function and reduce it to the most simplified form by applying various mathematical and arithmetic operations including addition, subtraction, multiplication, division, and many more.

The online calculator is comprised of two input tabs named Numerator and Denominator where the user inputs the data according to the desired function that needs to be solved. The calculator’s operation is very easy to understand and use, provided that the desired input function is valid.

How To Use the Rational Expression Calculator?

You can use the Rational Expression calculator by entering the numerator and denominator of the rational expression in the respective fields displayed on the calculator.

Here is a detailed explanation of how to use this calculator:

Step 1

Select the rational expression that needs to be rationalized.

Step 2

Identify the numerator and denominator in the rational expression.

Enter the numerator of the fraction in the Numerator tab.

Step 3

Now input the denominator in the Denominator tab.

Step 4

Once you have placed the numerator and the denominator, press the Simplify button.

Step 5

The result will be displayed in a new window. The new window shows two separate blocks. One block is named Input Interpretation, which displays the input in the form of the fraction that you have entered.

The second block is called Result. The resulting block has two options. You can either view the output generated by using the distributive method or the box method. Results displayed may vary in form depending upon the type of method selected. 

Moreover, the calculator also displays many forms of the expression just by clicking the option of More forms.

The rational expression calculator shows various forms of rationalized expression, each with different operations discussed below:

Option 1

Reduces the rational expression to obtain the lowest form.

Option 2

Performs mathematical operations such as multiplication, division, addition, and subtraction depending upon the function.

Option 3

Rationalizes the entire expression for the most optimized form of the rational expression.

Thus, it is a very easy-to-use calculator which displays all the simplified forms of rational expression.

How Does the Rational Expression Calculator Work?

The rational expression calculator works by using the factorization technique to rationalize the rational expressions and reduce the complex terms involved into simpler ones.

To solve these rational expressions manually, let us first discuss some important mathematical concepts and procedures involved.

What Is a Rational Expression?

A Rational Expression is a fraction wherein the numerator and denominator are in the form of algebraic polynomials. The denominator of a rational expression can never be equivalent to zero, therefore rational expression can also be defined as the ratio of two polynomials.

The standard form of the rational expression is given as:

\[ Rational Expression = \dfrac{ A (x) }{ B (x) } \]

A rational expression can involve either simple or complex polynomial functions. With the help of the Rational Expression Calculator, you can solve any expression in seconds with a detailed step-by-step solution that will not only enhance your understanding but will also help you solve complex problems.

An example of a rational expression is given below:

\[ \dfrac{ 6 x + 1 }{ 2 x + 1 } \] 

Any polynomial function is also considered a rational expression where the value of the denominator is given as 1. 

For instance, consider the following polynomial:

\[ 2 x^2 + 3 x  + 1 \]

If we write the above-mentioned polynomial as:

\[ \dfrac{ 2 x^2 + 3 x + 1 }{ 1 } \]

It will become a rational expression. Therefore, it can be stated that all the polynomial functions are also rational expressions.

When simplifying the rational expression, it is essential to segregate the common factors in the numerator and denominator and eliminate them. 

Operations Performed On the Rational Expressions

Here are the arithmetic operations that can be performed to solve and simplify the rational expressions:

  1. Addition
  2. Subtraction
  3. Multiplication
  4. Division

Addition

The two rational expressions can easily be added for simplification by following the steps given below:

  1. First, write all the terms separately in the form of a sum.
  2. Take the LCM of all the expressions to make the denominator common.
  3. Now add all the terms in the numerator of each expression over the common denominator. 
  4. Cancel the like terms with the opposite signs to get the simplified form of the expression.

Subtraction

Subtracting the two rational expressions is exactly similar to adding. Here are the steps that must be followed to simplify the rational expression:

  1. Write all the terms separately, such as in subtraction.
  2. Take the LCM for a common denominator.
  3. Subtract all the terms and cancel the like terms with the opposite signs.
  4. You can operate until the rational expression is reduced to the lowest form.

Multiplication

The process of Multiplying the rational expression is exactly similar to multiplying the numbers. Here are the steps to be followed:

  1. Multiply all the terms separately in the numerator and denominator.
  2. Apply the distributive property for multiplying the polynomials in the numerator and denominator.
  3. Add and subtract the terms accordingly to simplify the numerator and denominator. 
  4. Rewrite the expression in descending order to obtain a simplified form.

Division

To simplify two or more rational expressions using the division method, follow these steps:

  1. Write all the terms with the division sign.
  2. Take the reciprocal of expression and change the division sign into multiplication.
  3. Simplify the expressions by multiplying the terms in the numerator and denominator separately and then cancel the like terms with opposite signs.
  4. Reduce the expression to the lowest form.

Solved Examples

Here are some examples solved using the rational expression calculator:

Example 1

Consider the following rational Expression:

\[ \dfrac{x^2 – 6 x + 9 }{ ( x + 1) (x^2 – 1)} \]

Simplify the expression to its lowest form.

Solution

Use our calculator to simplify the rational expression given as:

\[ \dfrac{ x^2 – 6 x + 9 }{ ( x + 1) (x^2 – 1)} \]

Input the numerator and denominator in the respective tabs.

Numerator:

\[ x^2 – 6 x + 9 \]

Denominator:

\[ ( x + 1 )( x^2 -1 ) \]

Click the Simplify button to obtain the answer.

The result on the calculator is shown as:

\[ \dfrac{ ( x + 3 )^2}{ (x + 1)^2( x – 1 ) } \]

Click on more forms to view other simple forms of the expression with detailed steps.

Following are the steps shown with another simplified form of the rational expression:

\[ = \dfrac{x^2 – 6 x + 9 }{ ( x + 1) (x^2 – 1)} \]

Multiplying the denominator terms using distributive property gives us:

\[ = \dfrac { x^2 + 6x + 9}{x^3 + x^2 – x – 1} \]

Taking common terms out in both numerator and denominator:

\[ =  \dfrac{x( x + 6 ) + 9 }{ x ( x (x + 1) – 1 ) – 1} \]

Simplifying the expression gives us:

\[ = \dfrac{-3}{ x + 1} – \dfrac{ 2 }{ ( x + 1) ^2} + \dfrac { 4 }{ x – 1} \]

The final expression is given as:

\[ = \dfrac{ x^2 }{ x + 1 ) ( x^ – 1) } + \dfrac{ 6x }{(x + 1)( x^2 – 1)} + \dfrac{ 9 }{( x + 1)( x^2 – 1) } \]

Example 2

Simplify the following rational expression using the online rational expression calculator:

\[ \dfrac{ x^2 – 4 }{ x + 2 } \]

Solution

Use the calculator to simplify the rational expression to its lowest form.

Separate the numerator and denominator and input them into the respective field on the calculator.

The numerator is given as:

\[ x^2 – 4 \]

The denominator is given as:

 x + 2

The result is shown as follows:

 = x – 2 

Example 3

Simplify the following rational expression:

\[ \dfrac{ x^2 + 5x + 5 }{ x^3 + 7x + 35 } \]

Solution

Input the numerator and denominator in the calculator.

The Numerator is given as:

\[ x^2 + 5x + 5 \]

The Denominator is given as:

\[ x^3 + 7x + 35 \]

The result is given as:

\[ = \dfrac{ 5x }{ x^3 + 7x + 35} + \dfrac{ 5 }{ x^3 + 7x + 35} + \dfrac{ x^2 }{ x^3 + 7x + 35} \]

Another simplified form of the given rational expression with the stepwise solution is given as:

First, separate the common terms in the numerator and then in the denominator:

\[ = \dfrac{ x ( x + 5) + 5}{ x^3 + 7x + 35} \]

\[ = \dfrac{ x ( x + 5) + 5}{ x ( x^2 + 7) + 35 } \]

The final result is given as:

\[ = \dfrac{ x ( x + 5) + 5}{ x ( x^2 + 7) + 35 } \]

Therefore, using the calculator, you can simplify all kinds of rational expressions in the blink of an eye.

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