Master complex rational expressions in seconds with our Partial Fraction Decomposition Calculator. Whether you’re solving integrals, tackling Laplace transforms, or simplifying algebraic fractions, this powerful tool breaks problems into easy-to-handle steps—complete with clear explanations and accurate results every time. Perfect for students, educators, and professionals who want speed, precision, and understanding in one place.
Advanced mathematical tool for decomposing rational functions into simpler fractions
Calculating partial fraction decomposition...
Result: A/(x + 2) + B/(x + 3)
Result: A/(x - 1) + B/(x - 1)² + C/(x - 1)³
Result: A/x + (Bx + C)/(x² + x + 1)
Use standard mathematical notation: x^2 for x², * for multiplication, parentheses for grouping
The degree of the numerator must be less than the degree of the denominator
The denominator should be factorizable into linear and/or irreducible quadratic factors
Always verify your result by adding the partial fractions back together
This calculator is designed for educational purposes. While we strive for accuracy, please verify important calculations independently. The tool handles most common cases of partial fraction decomposition but may have limitations with very complex expressions.
Partial fraction decomposition is a cornerstone technique in algebra and calculus. If you’ve ever faced a rational expression, one polynomial divided by another, you know that simplifying it into smaller, more manageable “fractions” can make solving problems infinitely easier. A partial fraction decomposition calculator by CalcViva automates this process, breaking down complex fractions into simpler components you can use for integration, Laplace transforms, or solving differential equations. This guide explains the rules, the math, and how our calculator works, while integrating related key terms like rational function decomposition, proper vs. improper rational expressions, partial fraction expansion, and irreducible quadratic factors. By the end, you’ll not only know how to use the calculator but also understand the logic behind it—building both confidence and mathematical insight.
Using the calculator is straightforward:
Enter the numerator and denominator separately or paste the complete fraction.
Check the classification the tool will automatically determine if the fraction is proper or requires polynomial long division.
Factor the denominator the calculator applies symbolic factorization to identify linear and quadratic components.
Choose the method for simple linear factors, the cover-up method is fastest; otherwise, the coefficient comparison method is applied.
View step-by-step results including factoring, decomposition setup, solving for coefficients, and the final simplified form.
By automating these steps, the calculator saves significant time while maintaining accuracy, especially for complicated expressions that would take several minutes to solve manually.
A rational function is a fraction where the numerator and denominator are both polynomials. For example:
Before performing partial fraction decomposition, you must check whether the rational expression is proper or improper. A proper rational expression has a numerator degree strictly less than the denominator’s degree. If it’s improper, you must perform polynomial long division to rewrite it as a polynomial plus a proper fraction. This ensures the decomposition process works correctly.
| Term | Definition | Example |
|---|---|---|
| Proper Rational Expression | deg(numerator) < deg(denominator) | (x + 1) / (x² + 3x + 2) |
| Improper Rational Expression | deg(numerator) ≥ deg(denominator) | (x² + 4x + 3) / (x² + x + 1) |
| Polynomial Long Division | Divides polynomials to rewrite improper fractions | (x² + 4x + 3) ÷ (x² + x + 1) |
Understanding this classification is critical. The calculator automatically detects if division is needed before decomposition, ensuring accuracy.
The structure of the denominator dictates how you set up your decomposition. Mathematicians classify factors into four main types:
Distinct Linear Factors: Each factor is linear and unique, e.g., (x – 1)(x + 2).
Repeated Linear Factors: A factor appears with higher multiplicity, e.g., (x – 1)².
Irreducible Quadratic Factors: Quadratic terms that cannot be factored over the reals, e.g., x2+4x^2 + 4.
Repeated Quadratic Factors: An irreducible quadratic raised to a power, e.g., (x2+4)2(x^2 + 4)^2.
| Factor Type | Decomposition Form |
|---|---|
| Distinct Linear | A/(x – a) + B/(x – b) |
| Repeated Linear | A/(x – a) + B/(x – a)² |
| Irreducible Quadratic | (Ax + B)/(quadratic) |
| Repeated Quadratic | (Ax + B)/(quadratic) + (Cx + D)/(quadratic)² |
The calculator automatically factors the denominator and applies the correct decomposition rules. This systematic approach reduces human error and saves time, especially in cases involving irreducible quadratic factors or high-multiplicity roots.
Once the decomposition form is set, the next step is to solve for the unknown constants (A, B, C…). Two popular methods are:
Heaviside Cover-Up Method – Best for distinct linear factors. It allows you to find each constant by “covering up” the relevant factor and substituting a specific value for x. This method is fast and avoids solving large systems of equations.
Equating Coefficients Method – Works for all factor types, including irreducible quadratics. Here, you expand the decomposition, collect like terms, and equate coefficients of corresponding powers of x to form a solvable linear system.
The calculator uses symbolic algebra to apply both techniques appropriately, ensuring the decomposition is correct regardless of factor complexity. This is crucial for engineering and physics students who often deal with repeated quadratic factors in Laplace transforms.
Partial fraction decomposition is not just an academic exercise—it’s an indispensable tool in applied mathematics.
Integration: Many integrals involving rational functions become trivial after decomposition. For instance,
Differential Equations: Inverse Laplace transforms often require decomposing rational functions to match standard transform pairs.
Engineering & Physics: Transfer function simplification in control systems often involves repeated quadratic factors.
By understanding and applying partial fraction expansion, you bridge the gap between algebraic manipulation and real-world problem solving.
This calculator’s algorithm is built on well-established algebraic principles taught in university-level mathematics. Its methods align with respected academic resources like Paul’s Online Math Notes, Omni Calculator’s step-by-step solver, and Symbolab’s equation breakdowns. Each calculation is transparent, no hidden “black box” steps, and users can follow the reasoning just as they would on paper.
Moreover, this tool has been reviewed by mathematics educators with experience teaching calculus, algebra, and applied mathematics. Its accuracy and clarity make it suitable for students, teachers, engineers, and scientists. The focus is not only on delivering the correct answer but also on demonstrating the method, helping users build their own competence in partial fraction decomposition.
Even if you’re crunching numbers for school, business, or everyday life, our powerful online calculators help you get accurate results in seconds. From basic math to advanced financial tools, explore a growing collection designed to make your calculations faster, easier, and hassle-free.
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