Partial Fraction Decomposition Calculator / Solver

Partial Fraction Decomposition Calculator

Enter the rational function into the partial fractions calculator to decompose the partial fractions.

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Partial Fraction Decomposition Calculator

By using this partial fraction calculator, you can decompose complex rational expressions into the sum of simpler fractions. This partial fraction expansion calculator is useful for solving integrals, solving differential equations, or performing other mathematical operations that involve rational functions.

What Is Partial Fraction Decomposition?

“The process of transforming complex rational expressions into simpler fractions is called partial fraction decomposition”

It is also referred to as the partial fraction expansion and it is considered a valuable process when we are taking the antiderivatives of many rational functions.

Basic Principle:

In the partial fraction expansion, the fraction is given in the form S = p(x) / q(x) in which both the numerator and the denominator are univariate polynomials in the indeterminate x over a field.

What are the Types of Partial Fractions?

Type # 1: Unrepeated linear factor

= A ax + b

Type # 2: Repeated linear factor

= B (ax + b)^m , Where m ≥ 2

Type # 3: Unrepeated quadratic factor

= Cx + D (ax² + bx + c)

Type # 4: Repeated quadratic factor

= Ex + F (ax² + bx + c)ⁿ , Where n ≥ 2

How to Calculate Partial Fraction?

To calculate the partial fraction, express the rational function as a sum of simpler fractions with smaller denominators. This decomposition makes it easier to work by using online partial fraction decomposition calculator with steps. But here we have an example to make the concept easy.

Example:

Express the 5x + 10 (x + 1)(x + 6) as a partial fraction decomposition.

Solution:

Partial decomposition is given in the form:

5x + 10 (x + 1)(x + 6) = A (x + 1) + B (x + 6)

Write the first side as:

5x + 10 (x + 1)(x + 6) = (x+1) B + (x+6) A (x + 1)(x + 6)

The denominators are equal, so we require the equality of the numerators:

5x + 10 = (x+1) B + (x+6) A
5x + 10 = xB +B+ xA+6A
5x + 10 = xA+xB +6A + B

Collect up the like terms:

5x + 10 = x (A+B) +6A + B

The coefficients near the like terms should be equal, so the following system is obtained:

A + B = 5
6A + B = 10

For solving it, we consider A = 1, B = 4

Therefore, the answer is as follows:

5x + 10 (x + 1)(x + 6) = 1 (x + 1) + 4 (x + 6)

References:

From the source Newcastle University: Partial Fractions.

From the source Wikipedia: Partial fraction decomposition, Basic principles & reduction steps.