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Partial Fraction Decomposition Calculator

Decompose a rational expression (Ax + B) / ((x - r₁)(x - r₂)) into partial fractions.

Reviewed by Christopher FloiedPublished Updated

This free online partial fraction decomposition calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Results

C₁ (coefficient of 1/(x - r₁))

0

C₂ (coefficient of 1/(x - r₂))

1

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Partial Fraction Decomposition Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Partial Fraction Decomposition Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Partial Fraction Decomposition Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Partial Fraction Decomposition Calculator

The Partial Fraction Decomposition Calculator breaks down a rational expression with a linear numerator and two distinct linear factors in the denominator into a sum of simpler fractions. Partial fractions are essential in calculus for integrating rational functions, in differential equations for inverse Laplace transforms, and in signal processing for analyzing transfer functions. The idea is to reverse the process of adding fractions: instead of combining C₁/(x - r₁) + C₂/(x - r₂) into a single fraction, we decompose the combined form back into its simpler parts. Each resulting fraction is easier to integrate, differentiate, or analyze individually. The cover-up method (also called the Heaviside method) provides a quick way to find the coefficients.

The Math Behind It

Partial fraction decomposition expresses a proper rational function P(x)/Q(x) (where deg P < deg Q) as a sum of simpler fractions based on the factorization of Q(x). For distinct linear factors, (Ax+B)/((x-r₁)(x-r₂)) = C₁/(x-r₁) + C₂/(x-r₂). To find C₁, multiply both sides by (x - r₁) and set x = r₁: C₁ = (Ar₁+B)/(r₁-r₂). Similarly, C₂ = (Ar₂+B)/(r₂-r₁). This is the cover-up (Heaviside) method. For repeated linear factors (x-r)², the decomposition includes C₁/(x-r) + C₂/(x-r)². For irreducible quadratic factors (x²+bx+c), the numerator is linear: (Cx+D)/(x²+bx+c). If deg P ≥ deg Q, polynomial long division must be performed first to get a polynomial plus a proper fraction. Partial fractions are the key technique for integrating rational functions in calculus, as each term integrates to a logarithm, arctangent, or power function. In control theory and electrical engineering, partial fractions decompose transfer functions to identify individual pole contributions.

Formula Reference

Partial Fraction Form

(Ax+B)/((x-r₁)(x-r₂)) = C₁/(x-r₁) + C₂/(x-r₂)

Variables: C₁ = (Ar₁+B)/(r₁-r₂), C₂ = (Ar₂+B)/(r₂-r₁)

Worked Examples

Example 1: Simple decomposition

Decompose (3x + 5)/((x - 1)(x + 2))

Step 1:r₁ = 1, r₂ = -2, A = 3, B = 5
Step 2:C₁ = (3(1) + 5)/(1 - (-2)) = 8/3
Step 3:C₂ = (3(-2) + 5)/((-2) - 1) = (-1)/(-3) = 1/3
Step 4:Result: (8/3)/(x-1) + (1/3)/(x+2)

(8/3)/(x-1) + (1/3)/(x+2)

Common Mistakes & Tips

  • !Forgetting to perform polynomial long division first when the degree of the numerator is not less than the denominator
  • !Using the wrong form for repeated roots: (x-r)² requires A/(x-r) + B/(x-r)², not just A/(x-r)²
  • !Sign errors in the cover-up method when substituting negative roots
  • !Not fully factoring the denominator before setting up the decomposition

Related Concepts

Polynomial Division

Required when the numerator degree equals or exceeds the denominator degree

Factoring Trinomials

Factor the denominator into linear or irreducible quadratic factors

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Polynomial Division Calculator

Frequently Asked Questions

Why are partial fractions useful in calculus?

Each partial fraction term like C/(x-r) integrates to C·ln|x-r|, which is much simpler than trying to integrate the combined fraction directly. This technique converts difficult integrals into sums of elementary integrals.

What if the denominator has an irreducible quadratic factor?

For irreducible quadratic factors like (x² + 1), the corresponding partial fraction has a linear numerator: (Cx + D)/(x² + 1). This integrates to a combination of logarithm and arctangent functions.

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