# Partial Fraction Decomposition Calculator

## Find partial fractions step by step

This online calculator will find the partial fraction decomposition of the rational function, with steps shown.

Enter the numerator:

Enter the denominator:

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Partial Fraction Decomposition Calculator is a handy online tool that helps you decompose rational expressions into simpler fractions. Understanding the basics of partial fraction decomposition is critical when learning higher-level math topics.

## How to Use the Partial Fraction Decomposition Calculator?

• ### Input

Enter the numerator and denominator of a rational expression you wish to decompose.

• ### Calculation

Once you've entered the data, click the "Calculate" button.

• ### Result

The calculator will quickly process the expression and display the result of decomposition.

## What Is Partial Fraction Decomposition?

Partial fraction decomposition is a method used in algebra and calculus to decompose complex rational expressions into simpler fractions, making them easier to manipulate, especially during integration.

A rational expression (or a rational function) is a fraction in which the numerator and denominator are polynomials. Sometimes these expressions can be quite complex and difficult to work with.

The goal of partial fraction decomposition is to take a complex rational expression and decompose it into simpler fractions that are easier to work with.

Basic Concept

A rational expression has the following form:

$$R(x)=\frac{P(x)}{Q(x)},$$

where $P(x)$ and $Q(x)$ are polynomials, and the degree of $P(x)$ is less than the degree of $Q(x)$. If the degree of $P(x)$ is greater than or equal to $Q(x)$, then we can use polynomial division first and then decompose the resulting quotient.

The primary goal of partial fraction decomposition is to express $R(x)$ as a sum of simpler fractions.

Types of Fractions

• Proper Fraction

This is when the degree of the numerator is less than the degree of the denominator.

• Improper Fraction

This is when the degree of the numerator is greater than or equal to the degree of the denominator. In this case, divide the numerator by the denominator to get a polynomial plus a proper fraction, then decompose the proper fraction.

Steps to Decompose

1. Factor the Denominator: The first step is to factor the polynomial in the denominator.
2. Decompose the Fraction: Decompose the fraction depending on the factors of the denominator:

• For simple factors like $x-a$:

$$\frac{A}{x-a}$$
• For repeated factors like $(x-a)^n$:

$$\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\ldots+\frac{A_n}{(x-a)^n}$$
• For irreducible quadratic factors like $ax^2+bx+c$:

$$\frac{Ax+B}{ax^2+bx+c}$$
3. Solve for the Coefficients: Using various algebraic methods, determine the values of the constants.

For example, decompose $\frac{x^2+4x+3}{x^3-x^2-6x}$.

1. Factor the denominator:

$$x^3-x^2-6x=x(x-3)(x+2)$$
2. Decompose the fraction:

$$\frac{x^2+4x+3}{x^3-x^2-6x}=\frac{A}{x}+\frac{B}{x-3}+\frac{C}{x+2}$$
3. Bring to a common denominator and collect the like terms:

$$\frac{A}{x}+\frac{B}{x-3}+\frac{C}{x+2}=\frac{A(x-3)(x+2)+Bx(x+2)+Cx(x-3)}{x(x-3)(x+2)}=\frac{(A+B+C)x^{2}+(2B-A-3C)x-6A}{x(x-3)(x+2)}$$

This should equal to $\frac{x^2+4x+3}{x^3-x^2-6x}$:

$$\frac{x^2+4x+3}{x^3-x^2-6x}=\frac{(A+B+C)x^{2}+(2B-A-3C)x-6A}{x(x-3)(x+2)}$$

The denominators are equal, so the numerators should also be equal: $x^2+4x+3=(A+B+C)x^{2}+(2B-A-3C)x-6A$.

The coefficients near the like terms should be equal:

$$\begin{cases}A+B+C=1\\2B-A-3C=4\\-6A=3\end{cases}$$

In solving this system, we get that $A=-\frac{1}{2}$, $B=\frac{8}{5}$, $C=-\frac{1}{10}$.

Therefore, $\frac{x^2+4x+3}{x^3-x^2-6x}=-\frac{1}{2x}+\frac{8}{5(x-3)}-\frac{1}{10(x+2)}$.

## Why Choose Our Partial Fraction Decomposition Calculator?

• ### Accuracy and Correctness

Our calculator undergoes rigorous testing to ensure consistently correct results. You can rely on its accuracy.

• ### User-Friendly Design

We prioritize ease of use. The intuitive interface ensures that students and professionals can quickly navigate and obtain results without any hassle.

• ### Fast Computations

Our calculator delivers instant results, ensuring you don't have to wait unnecessarily.

• ### Educational Value

Alongside the result, the calculator provides an explanation to deepen your understanding of partial fraction decomposition.

### FAQ

#### What is the partial decomposition method?

The partial decomposition method, commonly referred to as partial fraction decomposition, is a technique in algebra used to decompose a complex rational expression into the sum of simpler fractions. This method simplifies the manipulation of rational expressions, especially during operations like integration in calculus. The goal is to express a given rational expression as a sum or difference of simpler rational expressions.

#### What is the Partial Fraction Decomposition?

The Partial Fraction Decomposition is a mathematical process that takes a complex rational expression and expresses it as a sum of simpler fractions. It is especially valuable in calculus when integrating rational expressions. This decomposition allows for easier algebraic and calculus-based manipulations, making complex problems more manageable.

#### What is the Partial Fraction Decomposition Calculator designed to do?

Our Partial Fraction Decomposition Calculator is designed to decompose complex rational expressions into simpler fractions, aiding in understanding and simplifying mathematical operations involving these expressions.

#### Can I use the calculator for complex rational expressions?

Our calculator is designed to handle a wide range of rational expressions, from simple to complex, and provide accurate decomposition results.