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Rational Expressions Calculator

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

To Calculate:

Simplify any Expressions:

Enter an Expression:

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Rational Expressions Calculator

Our online rational expressions calculator helps you factorize, simplify, and perform arithmetic operations on rational expressions. It can handle multiple operations at once and reduces expressions to their simplest form for easy understanding.

What Is a Rational Expression?

A rational expression is:

“A fraction where the numerator and/or denominator is an algebraic polynomial.”

In simpler terms, it is a fraction of polynomials. Even a polynomial can be considered a rational expression if written over 1:

Example:

$$ 2x^2 + 7x + 50 = \frac{2x^2 + 7x + 50}{1} $$

You can also use an online rational function calculator to simplify such expressions quickly.

Operations on Rational Expressions

Addition

Write all terms as a sum.
Find a common denominator (LCM).
Add the numerators and keep the denominator unchanged.
Cancel opposite terms and simplify the rest.

Subtraction

Write all terms as a subtraction.
Find a common denominator (LCM).
Subtract the numerators, keeping the denominator unchanged.
Simplify by canceling common terms.

Multiplication

Multiply the numerators together and the denominators together.
Use the distributive property if necessary.
Combine like terms and write in descending powers.

Division

Change the division to multiplication by flipping the numerator and denominator of all but the first term.
Follow multiplication rules afterward.

How to Simplify Rational Expressions

Example:

Simplify:

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

Step 1: Factor numerator and denominator:

$$ x^2 - 6x + 9 = (x - 3)^2 $$

$$ x^2 - 1 = (x + 1)(x - 1) $$

Step 2: Rewrite the rational expression:

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

Step 3: Simplify repeated terms:

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

This is the simplified form.

How the Rational Expressions Calculator Works

Our calculator can simplify, add, subtract, multiply, and divide rational expressions quickly. Here’s how it works:

Input Options:

Reduce terms of a rational expression.
Add, subtract, multiply, or divide two or three expressions.
Simplify any rational expression.

Steps for Option 1 (Reduce Terms)

Enter numerator and denominator.
Click "Calculate".

Steps for Option 2 (Arithmetic Operations)

Choose whether to operate on 2 or 3 expressions.
Enter numerator and denominator for each expression.
Click "Calculate".

Steps for Option 3 (Simplify Expression)

Enter the entire rational expression.
Click "Calculate".

Output:

Option 1: Step-by-step reduction to simplest form.
Option 2: Result of addition, subtraction, multiplication, or division.
Option 3: Fully simplified rational expression.

FAQs

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

Proper: Degree of numerator < degree of denominator.
Improper: Degree of numerator ≥ degree of denominator.

What is the degree of a polynomial?

The highest power of the variable in a polynomial.

Example:

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

Degree = 9 (highest exponent)

What is a monomial?

A polynomial with only one term.

Example: 5z, 4x, 65y

Conclusion

Simplifying rational expressions reduces complexity in algebraic problems. Professionals in mathematics, data science, engineering, and physics often use online rational function calculators for fast, accurate results.

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