Decomposing Functions into Partial Fractions: A Comprehensive Guide

Have you ever encountered a complex fraction that seems difficult to solve or manipulate? One powerful method in calculus and algebra is decomposing functions into partial fractions. This article will introduce you to partial fractions, guide you through the steps of decomposition, and provide a detailed example to help you understand the process better.

What are Partial Fractions?

Partial fractions refer to the process of breaking down a complex rational function (a fraction where both the numerator and denominator are polynomials) into simpler fractions called partial fractions. These simpler fractions are much easier to work with, especially when it comes to integration or solving differential equations.

Techniques for Decomposing Functions into Partial Fractions

Ensure Proper Fraction

First, check if the degree of the numerator is less than the degree of the denominator. If not, perform long division to rewrite it as a proper fraction plus a polynomial.

Factor the Denominator

Factor the denominator of the rational function if possible. The form of the decomposition depends on the type of factors: linear or quadratic.

Set Up Partial Fractions

Break the rational function into simpler fractions based on the factors of the denominator. The setup depends on the type of factors:

Linear Factors e.g. x - a:

[frac{A}{x - a}]

Repeated Linear Factors e.g. (x - a^n):

[frac{A_1}{x - a} frac{A_2}{x - a^2} cdots frac{A_n}{x - a^n}]

Irreducible Quadratic Factors e.g. (x^2 - bx - c):

[frac{Ax B}{x^2 - bx - c}]

Repeated Quadratic Factors e.g. ((x^2 - bx - c)^n):

[frac{A_1x B_1}{x^2 - bx - c} frac{A_2x B_2}{(x^2 - bx - c)^2} cdots frac{A_nx B_n}{(x^2 - bx - c)^n}]

Multiply by the Denominator

Multiply both sides of the equation by the common denominator to eliminate the fractions.

Solve for Coefficients

Expand the equation and collect like terms. Then, equate the coefficients of corresponding powers of x to form a system of linear equations. Solve this system to find the unknown coefficients.

Example: Decompose (frac{3x 5}{x^2 - x - 2})

Let's break down the process with an example:

The denominator is already factored: (x - 1) and (x 2).

Set up the partial fractions:

[frac{3x 5}{(x 1)(x - 2)} frac{A}{x - 1} frac{B}{x 2}]

Multiply both sides by the common denominator:

3x 5 A(x 2) B(x - 1)

Expand and collect like terms:

3x 5 Ax 2A Bx - B (A B)x (2A - B)

Equate the coefficients:

x terms: (A B 3)

Constant terms: (2A - B 5)

Solve the system of equations:

From (A - B 3), we get (B 3 - A).

Substitute into (2A - B 5):

2A - (3 - A) 5 rarr; 2A - 3 A 5 rarr; 3A - 3 5 rarr; 3A 8 rarr; A (frac{8}{3})

Substitute (A frac{8}{3}):

B 3 - (frac{8}{3}) (frac{9}{3}) - (frac{8}{3}) (frac{1}{3})

Therefore, the partial fraction decomposition is:

[frac{3x 5}{(x 1)(x - 2)} frac{(frac{8}{3})}{x 1} frac{(frac{1}{3})}{x - 2}]

You can leave the fractions as they are or combine the constants with the numerators.

Summary

Decomposing a function into partial fractions is a method that simplifies complex rational functions into more manageable pieces. By following the steps outlined in this guide, you can efficiently handle various types of rational functions. Whether you are working with linear or quadratic factors, repeated terms, or a mix of different types, the process remains consistent and logical.