# Partial Fraction Decomposition Calculator

> Decompose rational expressions into partial fractions with step-by-step work for calculus and algebra

**Category:** Math
**Keywords:** partial fractions, rational expressions, decomposition, calculus, algebra, integration
**URL:** https://complete.tools/partial-fraction-decomposition

## How partial fraction decomposition works

Partial fraction decomposition reverses the process of combining fractions. Given a rational expression where the degree of the numerator is strictly less than the degree of the denominator, you can always write it as a sum of simpler fractions.

The general procedure is:

1. Factor the denominator completely into linear and irreducible quadratic factors.
2. Write a template where each factor contributes one or more unknown-numerator terms.
3. Multiply both sides by the full denominator to clear all fractions.
4. Solve for the unknown constants by substituting the roots of each factor or by comparing coefficients of like powers of x.

For example, 1 / ((x+1)(x-2)) decomposes as A/(x+1) + B/(x-2). Clearing denominators gives 1 = A(x-2) + B(x+1). Substituting x = -1 isolates A = -1/3, and substituting x = 2 isolates B = 1/3.

Key rules for the template:
- A distinct linear factor (x+a) gets a constant A in the numerator.
- A repeated linear factor (x+a)^n gets n terms: one for each power up to n.
- An irreducible quadratic factor (x^2+bx+c) gets a linear numerator Bx+C.

## How to use

1. Select the decomposition form that matches your expression from the dropdown.
2. Enter the value of N (the numerator constant or coefficient).
3. For forms with a linear numerator (Form 2), also enter M.
4. Enter the parameter a (the offset in the first linear factor).
5. For forms with two factors, enter b (the offset in the second factor or the x-coefficient of the quadratic).
6. For Form 4, also enter c (the constant term of the quadratic).
7. Click Decompose to see the full step-by-step solution.
8. Review each numbered step, the coefficients found, and the verification check.
9. Click Start Over to clear results and try a different expression.

## When to use partial fractions

Partial fraction decomposition is most frequently used as a technique for integrating rational functions. After decomposing, each simpler term can be integrated using basic antiderivative rules:

- A constant over a linear factor A/(x+a) integrates to A times ln|x+a| plus C.
- A constant over a squared linear factor B/(x+a)^2 integrates to negative B/(x+a) plus C.
- A linear numerator over an irreducible quadratic uses arctangent and logarithm formulas after completing the square.

Beyond integration, partial fractions appear in Laplace transform inversion, control systems analysis, and solving certain recurrence relations in discrete mathematics.

## FAQs

**Q:** What does it mean for a quadratic to be irreducible?


**A:** A quadratic x^2+bx+c is irreducible over the real numbers when its discriminant b^2-4c is negative, meaning it has no real roots and cannot be factored into two real linear factors. The expression 1/((x+1)(x^2+4)) illustrates this since x^2+4 has no real roots.


**Q:** What if the degree of the numerator is not less than the degree of the denominator?


**A:** Partial fraction decomposition only applies when the numerator degree is strictly less than the denominator degree. If the degrees are equal or the numerator has higher degree, you must first perform polynomial long division to get a polynomial remainder plus a proper rational expression, then decompose the proper part.


**Q:** Why does a repeated factor need multiple terms?


**A:** A single term A/(x+a)^2 cannot represent all rational expressions with that denominator. Including both A/(x+a) and B/(x+a)^2 provides enough degrees of freedom to match any numerator of degree less than 2. Without the first-power term the system of equations would be underdetermined.


**Q:** Can I use this for complex roots?


**A:** This tool handles real coefficients only. When the quadratic factor has a negative discriminant and no real roots it is treated as irreducible and assigned a linear numerator Bx+C. Working with complex roots directly is equivalent but requires complex arithmetic. The real-coefficient approach using irreducible quadratics is the standard method for real-valued calculus.


**Q:** How do I verify the decomposition is correct?


**A:** Multiply each partial fraction term by the common denominator and add the results together. The sum should equal the original numerator. This calculator performs that check automatically and displays the verification expression below the solution steps.

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