Partial Fraction Decomposition Calculator

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:

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.

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