How to Express 2/((1 x)(13x)) in Partial Fractions Using Heaviside’s Coverup Method

In algebra, partial fraction decomposition is a technique used to break down complex rational functions into simpler, more manageable fractions. This process is crucial in various mathematical and engineering applications. One such technique is Heaviside’s coverup method, which simplifies the decomposition procedure significantly. This article will guide you through the process of expressing 2/((1 x)(13x)) in partial fractions using Heaviside’s coverup method. We will also provide a detailed explanation of the process and include a verification step to ensure our solution is correct.

Introduction to Partial Fractions

A rational function is a fraction where both the numerator and the denominator are polynomials. The process of decomposing a rational function into partial fractions aims to simplify it into a sum of simple rational expressions that are easier to integrate or analyze. Heaviside’s coverup method is a shortcut technique for finding the coefficients of these partial fractions when the denominator can be factored into distinct linear factors.

Understanding Heaviside’s Coverup Method

Heaviside’s coverup method is a popular technique for decomposing partial fractions. It involves setting the variable equal to each root of the denominator and then solving for the coefficients. Here's a step-by-step guide to using this method:

Start with a rational function. In our case: 2/((1 x)(13x)).

Assume the partial fraction decomposition has the form:

A1 x B13x2(1 x)(13x)]

Solve for the coefficients A and B using the roots of the denominator.

Solving for A and B

To find the values of A and B, we use the coverup method as follows:

Set 1 x0. This gives us x-1. Multiply through by (1 x):

213[-1]A]

Therefore, A-1.

Set 13x0. This gives us x0. Multiply through by (13x):

21[0-13]B]

Therefore, B3.

Partial Fraction Decomposition

With A-1 and B3, the partial fraction decomposition of 2/((1 x)(13x)) is:

-11 x 313x

Thus, we have:

2(1 x)(13x)-11 x 313x

Verification

To ensure the correctness of our decomposition, we can verify that the partial fractions add up to the original rational function. Let's check:

-113x31x(1 x)(13x)-1-3x33x(1 x)(13x)2(1 x)(13x)

This confirms that our partial fraction decomposition is correct:

2(1 x)(13x)-11 x 313x

By following these steps, you can easily express complex rational functions in partial fractions using Heaviside’s coverup method. This not only simplifies the function but also makes it easier to perform various mathematical operations on it.