(2.1)(2.1)

Classroom Tips and Techniques: The Partial Fraction Decomposition

Robert J. LopezEmeritus Professor of Mathematics and Maple Fellow

© Maplesoft, a division of Waterloo Maple Inc., 2006

Introduction

Students typically meet the algebraic technique of partial fraction decomposition in their course in integral calculus. It is presented as part of the methodology for integrating rational functions, but it isreally nothing more than an algebraic process independent of its use in integration. Indeed, the second time students meet partial fractions is in a differential equations course, one where the Laplace transform appears.

The classic technique for inverting Laplace transforms is to apply pattern recognition to the terms produced by a partial fraction decomposition. When this process is implemented in some engineeringclasses, the decomposition must be given strictly in terms of linear factors, even if these factors are complex.

Maple provides tools for studying and implementing the partial fraction decomposition, and in this column we describe how both objectives can be met.

Combining Fractions

To add the fractions in

(2.2)(2.2)

(3.1)(3.1)

(2.3)(2.3)

(3.2)(3.2)

(2.4)(2.4)

either the simplify or normal commands can be used, as we see from

and

The normal command is more specifically designed to combine fractions, and therefore supports an additional parameter, expanded, which causes the factors in the denominator to be multiplied out. This is illustrated by

Both the simplify and normal commands are accessible from the pop-up menu that appears by right-clicking on the sum in (1.1). However, the expanded option for normal is not available in this point-and-click format.

The Partial Fraction Decomposition

The process of reversing the addition of fractions is called "partial fraction decomposition." To decompose the fraction

use the Maple command

(4.1.4)(4.1.4)

(4.1.2)(4.1.2)

(4.1.3)(4.1.3)

(4.1.1)(4.1.1)

(4.1.5)(4.1.5)

Conversion to "partial fraction" form is also available in the pop-up menu under the headingConversions.

Mechanics of the Decomposition

The Long-Division Step

Since the degree of the numerator in is greater than the degree of the denominator, a long

division precedes the construction of the decomposition. This step is implemented in Maple via

to obtain the dividend, and

to obtain the remainder. Both these commands are provided in a Task Template, obtained from the Tools/Tasks/Browse/Polynomials menu under the Polynomial Division Quotient and Remainder option.

Incidentally, both the dividend and the remainder can be computed with the quo command, using the syntax

The remainder has been assigned to the variable , as we see from

In either event, the actual target of the decomposition is the fraction

The factored form of the denominator can be obtained with the command

(4.1.6)(4.1.6)

or with the pop-up menu menu option Factor.

Pedagogical IssuesThe Template Fractions

It isn't enough to show students how to get Maple to provide a partial fraction decomposition. Most instructors would want students to have some insight into the computational details of the algebra used to evince the result.

To this end, the first thing the student needs to know is the set of template-fractions given in Table 1.

(4.2.1.1) (4.2.1.2)

(4.2.1.3) (4.2.1.4)

Table 1 Template fractions for the decomposition of

Thinking back to when I first learned the "algorithm" as a student, I remember making the association that fractions arising from linear factors took a simple numerator, namely, a constant, but factors arising from quadratic factors took a "messier" or more complex numerator. Somewhere between learning this as a student and articulating it for students, I formulated the "rules"

simple - simplemessy - messy

Watch out for the repeats!

Few students I shared this with willingly acknowledged the irony of the repetitions, but no student failed to grasp the concepts behind this terse summary of how to formulate the template fractions in a partial fraction decomposition.

An Identity in x

There are several computational techniques for determining the coefficients in the template

(4.2.2.3)(4.2.2.3)

(4.2.2.2)(4.2.2.2)

(4.2.2.1)(4.2.2.1)

fractions. They all hinge on the recognition that the sum of these fractions must identically equal the fraction being decomposed.

Working by hand, students can be taught to formulate enough equations by substituting "smart" numbers in for in the identity

where the left side is the fraction being decomposed and the right side is the sum of template fractions. "Smart" numbers are values of for which one or more terms vanish, making the resulting equations simpler. Incidentally, I observed over the years that irreducible quadratic factors gave students the hardest time if they were dependent solely on this technique.

Alternatively, the right side can be combined over a common denominator, and the numerator equated to . If that numerator is multiplied out, the equations resulting from equating coefficients of like powers determine the unknown coefficients.

The most straightforward Maple approach is to determine the unknown coefficients with

and to insert these values into the template fractions with

1

To implement the algebra at a more fundamental level, add the template fractions and select the resulting numerator. In Maple, we get

(4.2.2.5)(4.2.2.5)

(5.2)(5.2)

(5.3)(5.3)

(5.1)(5.1)

(4.2.2.4)(4.2.2.4)

Equating coefficients of like powers in this polynomial and leads to the equations

The solution of these equations is again

Decomposition to Linear Factors

The denominator in contains a reducible quadratic factor. Indeed, the quadratic factor

is reducible since its zeros are

Hence, in terms of linear factors, the partial fraction decomposition of (4.1.5) is given in Maple as

(5.5)(5.5)

(5.7)(5.7)

(5.6)(5.6)

(5.4)(5.4)

where the sums are over the zeros of (5.1). A more convenient representation of this

decomposition might be

where the zeros are explicitly given.

This process also works over the complex field, as we see when we decompose

to

As before, we can express the sum explicitly in terms of radicals, obtaining

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