Partial Fraction Decomposition of Rational Functions: A Comprehensive Guide

Partial fraction decomposition is a powerful technique in algebra used to simplify rational functions, especially when integrating or solving complex equations. This guide will walk you through the process of decomposing a rational function into simpler fractions, using both algebraic and calculus concepts.

Understanding Rational Functions and Partial Fractions

A rational function is any function that can be written as the ratio of two polynomials. For example, x^2 / (x^2 - 3x 2) is a rational function. Decomposing this function into partial fractions allows us to break it down into simpler components that are easier to handle.

Step-by-Step Guide to Partial Fraction Decomposition

Let's take the function x^2 / (x^2 - 3x 2) as our example and decompose it into partial fractions.

Step 1: Factor the Denominator

First, we factor the denominator:

x^2 - 3x 2 (x - 2)(x - 1)

Step 2: Set Up the Partial Fraction Decomposition

We express the original function as the sum of partial fractions:

x^2 / (x-2)(x-1) A/(x-2) B/(x-1)

Step 3: Combine and Simplify

Multiply both sides of the equation by the denominator (x-2)(x-1):

x^2 A(x-1) B(x-2)

Expand and combine like terms:

x^2 Ax - A Bx - 2B (A B)x - (A 2B)

Step 4: Equate Coefficients

We now set up a system of equations by equating the coefficients of x and the constant terms:

Coefficients of x: A B 1 Constants: -A - 2B -2

Step 5: Solve the System of Equations

From the second equation, solve for A:

-A - 2B -2 implies A 2 - 2B

Substitute A 2 - 2B into the first equation:

(2 - 2B) B 1 implies 2 - B 1 implies B 1

Substitute B 1 back into A 2 - 2B:

A 2 - 2(1) 0

Final Result

The partial fraction decomposition is:

x^2 / (x-2)(x-1) 2/(x-2) - 1/(x-1)

Decomposing Rational Functions with Higher Powers

For rational functions with higher powers in the denominator, the process is similar, but we need to ensure that the numerators are polynomials of lower degree. For example:

Decompose x^2 / (x-1)(x-2):

Factor the denominator: x^2 - 3x 2 (x-1)(x-2) Set up the partial fraction decomposition: x^2 / (x-1)(x-2) A/(x-1) B/(x-2) Multiply by the denominator: x^2 A(x-2) B(x-1) Combine like terms: x^2 Ax - 2A Bx - B Equate coefficients: Coefficients of x: A B 1 Constants: -2A - B -2 Solve the system of equations: From the second equation: -2A - B -2 implies A 1 B/2 Substitute A 1 B/2 into the first equation: 1 B/2 B 1 implies B/2 -1 implies B -2 Substitute B -2 back into A 1 B/2: A 1 - 1 0 The partial fraction decomposition is: x^2 / (x-1)(x-2) 0/(x-1) - 2/(x-2) or -2/(x-2)

Conclusion

Partial fraction decomposition is a valuable tool for simplifying complex rational functions. By breaking down the function into simpler components, we can more easily integrate, differentiate, or solve equations. This guide has provided a step-by-step approach to decomposing a rational function into partial fractions, suitable for both algebraic and calculus applications.

For a deeper understanding, you can explore examples and exercises involving higher degree polynomials and repeated factors in the denominator. Practice is essential to master this technique.