Partial Fraction Decomposition Calculator
* Note: This standard web tool supports decomposing proper fractions with up to quadratic (degree 2) denominators. Enter standard algebraic notation (e.g., 2x^2 - 5x + 3).
Partial fraction decomposition turns one complicated rational function into a sum of simpler fractions you can actually work with. Whether you are integrating in calculus, solving differential equations, or applying inverse Laplace transforms, this free partial fraction decomposition calculator gives you instant, accurate results.
Quick Definition: Partial fraction decomposition is an algebra technique that rewrites a single rational function – a fraction with polynomials in the numerator and denominator – as a sum of simpler fractions. Each simpler fraction has one of the denominator’s factors as its own denominator. This makes integration and equation-solving far more manageable.
Understanding Partial Fraction Decomposition
A rational function like (x + 7) / (x^2 + 3x) looks clean on paper. In practice, it blocks you every time you try to integrate it or apply a Laplace transform. You cannot integrate that expression directly with a standard formula.
Partial fractions fix this. They decompose the original expression into smaller pieces you CAN integrate one by one.
Real-world scenario: An electrical engineering student is working on a control systems problem. The transfer function has a complex rational expression in the denominator. Without decomposing it into distinct linear factors, applying the inverse Laplace transform is nearly impossible. This tool solves that in seconds.
Partial fraction decomposition also appears in:
- Calculus – breaking down integrands before integrating
- Differential equations – isolating terms to find particular solutions
- Control theory – analyzing system responses via Laplace transforms
The Math Behind Partial Fraction Decomposition
The core algorithm follows a fixed set of steps. Understanding these helps you verify the tool’s output and catch any input errors.
Step 1 – Check if the fraction is proper. The degree of the numerator must be less than the degree of the denominator. If it is not, you must perform polynomial long division first to get a proper fraction before decomposing.
Step 2 – Fully factor the denominator. Every denominator factors into one of three types:
- Distinct linear factors – e.g., (x)(x + 3)
- Repeated linear factors – e.g., (x – 2)^2
- Irreducible quadratic factors – e.g., (x^2 + 4), where the discriminant is negative and it has no real roots
Step 3 – Set up the partial fraction template.
| Denominator Factor Type | Partial Fraction Form |
|---|---|
| Distinct linear: (ax + b) | A / (ax + b) |
| Repeated linear: (ax + b)^2 | A / (ax + b) + B / (ax + b)^2 |
| Irreducible quadratic: (ax^2 + bx + c) | (Ax + B) / (ax^2 + bx + c) |
Step 4 – Multiply both sides by the common denominator. This clears all fractions and gives you a polynomial equation.
Step 5 – Solve for the unknown constants (A, B, C…). Use two methods:
- Substitution (cover-up method / Heaviside method) – plug in the roots of each factor to isolate one constant at a time
- Equating coefficients – match the coefficients of like powers on both sides to build a system of linear equations, then solve
Step 6 – Write the final decomposed form. Replace the original rational expression with the sum of simpler fractions you found.
According to MIT OpenCourseWare’s calculus resources, partial fraction expansion is one of the most tested integration techniques in undergraduate mathematics.
Interpreting the Decomposition Result
Once the tool displays the decomposition result, here is what to do with it:
If you are integrating: Each simpler fraction now matches a standard integral form. A term like A / (x + b) integrates directly to A x ln|x + b| + C. An irreducible quadratic factor term integrates using the arctan formula.
If you are applying inverse Laplace transforms: Each partial fraction maps to a known Laplace pair in standard tables, letting you write the time-domain solution immediately.
If you get an error or unexpected result: Double-check that:
- Your fraction is proper (degree of numerator is less than degree of denominator)
- You entered the denominator in standard algebraic notation (e.g., x^2 – 5x + 3)
- The denominator factors correctly and does not produce complex cases beyond degree 2
Key Features of This Partial Fraction Calculator
- Instant decomposition – results appear immediately after you click “Decompose Fraction”
- Supports all three factor types – distinct linear factors, repeated linear factors, and irreducible quadratic factors
- Standard algebraic notation input – enter polynomials exactly as you would write them (e.g., 2x^2 – 5x + 3)
- Works for calculus, algebra, and differential equations – one solver covers multiple use cases
- Print and share buttons – save or send your result with one click
- 100% free – no sign-up, no paywall, no data stored on any server
How to Use the Partial Fraction Decomposition Calculator
Follow these exact steps based on the tool interface:
Step 1 – Enter the Numerator N(x). Click the top input field labeled “Numerator N(x)”. Type your numerator polynomial using standard notation. Example: x + 7
Step 2 – Enter the Denominator D(x). Click the second input field labeled “Denominator D(x)”. Type your denominator polynomial. Example: x^2 + 3x
Use these notation rules:
- Write exponents as
x^2,x^3, etc. - Use standard operators:
+,-,xfor multiplication where needed - Do not use division symbols inside the fields
Step 3 – Click “Decompose Fraction”. Hit the large purple button. The tool instantly processes your rational function and displays the full decomposition result in the output box below.
Step 4 – Read the Decomposition Result. The result panel shows your original rational expression rewritten as a sum of simpler fractions with all constants (A, B, C) solved and substituted.
Step 5 – Print or Share. Use the “Print Result” button to save a clean copy, or click “Share” to send the result directly.
Note: This tool supports proper fractions with denominators up to quadratic (degree 2). If your denominator is degree 3 or higher, factor it manually first, then run each pair of factors through the tool. For more math tools, visit our Math Calculators collection.
Quick Reference: Partial Fraction Templates by Factor Type
| Factor Type | Example Denominator | Partial Fraction Template |
|---|---|---|
| Distinct linear factors | x(x + 3) | A/x + B/(x + 3) |
| Repeated linear factors | (x – 2)^2 | A/(x – 2) + B/(x – 2)^2 |
| Irreducible quadratic | x(x^2 + 4) | A/x + (Bx + C)/(x^2 + 4) |
| Two distinct linears | (x + 1)(x – 5) | A/(x + 1) + B/(x – 5) |
| Linear + repeated | x(x + 1)^2 | A/x + B/(x + 1) + C/(x + 1)^2 |
Accuracy and Trust Guarantee
This partial fraction calculator uses a verified algebraic algorithm that follows the same decomposition logic taught in university calculus and algebra courses. The tool:
- Applies the standard cover-up (Heaviside) method for distinct linear factors
- Uses coefficient-matching for repeated roots and irreducible quadratic cases
- Runs entirely in your browser – no data is sent to or saved on any server
- Is 100% free with no account required
- Stays updated to handle standard algebraic notation without encoding errors
Frequently Asked Questions (FAQs)
What is the difference between partial fraction decomposition and partial fraction expansion?
They are the same technique. Partial fraction expansion is simply another name for partial fraction decomposition – both refer to rewriting a rational function as a sum of simpler fractions with factored denominators.
Can I use this calculator for inverse Laplace transforms?
Yes. Partial fraction decomposition is a standard step before applying inverse Laplace transforms. Decompose your rational expression here, then match each resulting term to a known Laplace pair from a standard table to find the time-domain function.
What if my numerator degree equals or exceeds my denominator degree?
Your fraction is improper. You must perform polynomial long division first to extract a whole polynomial plus a proper remainder fraction. Then run only the proper remainder fraction through this calculator.
Does this tool handle repeated linear factors?
Yes. The tool supports denominators with repeated linear factors up to quadratic degree (degree 2). Enter the full denominator as written and the calculator correctly sets up and solves for all required constants.