Partial Fraction Decomposition Calculator Online

Enter a rational function of the form P(x) / Q(x):

✅ Partial Fraction Decomposition Calculator – Step-by-Step Rational Function Solver

The Partial Fraction Decomposition Calculator on CalcIreland is a free, easy-to-use online tool that breaks down any rational function into simpler fractional components. Whether you’re a student, teacher, engineering professional, or someone studying algebra and calculus, this tool helps you quickly decompose expressions of the form P(x) / Q(x) into partial fractions with accurate, step-by-step results.

Partial fraction decomposition is an essential concept in integration, calculus, algebra, Laplace transforms, and differential equations. Instead of solving manually—which can be time-consuming and error-prone—this calculator performs the entire process instantly and provides clean, simple fractions.

For more algebra tools, equation solvers, and mathematics utilities, explore our full Mathematics & Science Tools section.

✅ What Is Partial Fraction Decomposition?

Partial Fraction Decomposition is the process of expressing a complex rational function as a sum of simpler fractions.
For example:

$2x+3×2−1=Ax−1+Bx+1\frac{2x + 3}{x^2 – 1} = \frac{A}{x-1} + \frac{B}{x+1}$

This technique is widely used in:

Integration in calculus
Solving differential equations
Laplace transforms
Algebraic simplification
Engineering and signal processing
Calculus coursework and exams

Decomposing rational expressions manually requires factoring, comparing coefficients, solving systems of equations, and identifying denominator types. This calculator does all of that automatically.

You can also go through - Taylor Series Calculator

✅ How to Use the Partial Fraction Decomposition Calculator

Follow these simple steps:

1. Enter your rational function

Type your expression into the input box in the format:

(2x + 3) / (x^2 - 1)
(3x + 5) / (x^2 + 2x - 8)
(x + 7) / (x^2 - 4)

The calculator accepts linear numerators and quadratic denominators.

2. Click the “Calculate” button

The tool instantly analyzes your function, factors the denominator, and solves for constants A, B, C, etc., depending on the structure.

3. View the result

You’ll get a decomposition in the form:

$Ax−r1+Bx−r2\frac{A}{x – r_1} + \frac{B}{x – r_2}$

If applicable, the calculator also handles repeated roots or simplified result forms.

4. Use the result for further work

You can copy the output for:

Integration problems
Calculus assignments
Laplace transform solutions
Engineering computations
Exam preparation

✅ How the Partial Fraction Decomposition Calculator Works

This tool uses algebraic algorithms to:

✔ Step 1: Parse the numerator and denominator

It reads functions in standard polynomial form.

✔ Step 2: Factor the denominator

The calculator automatically factors:

Quadratics: $ax^2 + bx + c$
Linear factors
Distinct real roots
Repeated roots (if enabled)

✔ Step 3: Set up the partial fraction format

For example:

$2x+3(x−1)(x+1)=Ax−1+Bx+1\frac{2x + 3}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$

✔ Step 4: Solve the system of equations

The tool equates coefficients and calculates constants A, B, C, etc.

✔ Step 5: Generate the final simplified expression

Clean, readable, fraction-based output is displayed.

All steps are handled automatically with high accuracy.

$partial fraction decomposition calculator$

✅ Why Use This Partial Fraction Decomposition Calculator?

✔ 1. Fast and accurate

Manual decomposition can take several minutes.
With this tool, the result appears instantly.

✔ 2. Ideal for calculus and integration

Most integration problems involving rational functions require partial fractions.
This calculator makes your workflow much easier.

✔ 3. Perfect for students, teachers, and engineers

Useful for:

Leaving Cert maths students
University calculus
Engineering applications
Differential equation solving
Laplace transform computations
Competitive exam preparation

✔ 4. Supports Ireland-specific education needs

Optimized for students studying in Irish universities, colleges, and secondary schools following Irish maths curricula.

✔ 5. No installation needed

Just enter your function and get results—fast, simple, free.

✔ 6. Reduces calculation mistakes

Eliminates sign errors, incorrect factoring, and coefficient mistakes.

✅ FAQ: Partial Fraction Decomposition Calculator

1. What types of expressions can this calculator solve?

It supports rational functions with linear numerators and quadratic or linear denominators.
Example:
$2x + 3) / (x^2 – 1)$

2. Can this help with integration?

Yes. This tool is designed to simplify rational functions into integrable forms.

3. Do I need to factor the denominator manually?

No. The calculator automatically factors the denominator for you.

4. Does it support repeated roots?

Yes, repeated linear factors can be decomposed into multiple terms (if the option is enabled).

5. Is this calculator free to use?

Absolutely. All tools on CalcIreland are 100% free.

Partial Fraction Decomposition Calculator