Classroom Tips and Techniques: The Partial-Fraction Decomposition Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft 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 is really 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 engineering classes, 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 Table 1 examines how Maple's Context Menu can be invoked to sum two algebraic fractions. Define the fractions to be added Write the sum of two fractions. Context Menu: Assign to a Name_f assign to a name f Simplify Write the name f. Context Menu: Evaluate and Display Inline Context Menu: Simplify_Simplify simplify Normalize
Write the name f. Context Menu: Evaluate and Display Inline Context Menu: Normal normal Write the name f. Context Menu: Evaluate and Display Inline Context Menu: Simplify_Normalize Expanded Normalize but expand the denominator normalize expanded Table 1 Maple's syntax-free tools for adding two algebraic fractions The Partial-Fraction Decomposition The process of reversing the addition of algebraic fractions is called the "partial-fraction decomposition." Table 2 shows how to obtain a partial-fraction decomposition using syntax-free methods. Context Menu: Conversions_Partial Fractions_x convert to partial fractions in x Table 2 Partial-fraction decomposition via the Context Menu In Table 2, the rational function converted to partial fractions is the form of f appearing at the end of Table 1. The partial-fraction decomposition of f is the sum of three terms, whereas f itself is the sum of two fractions. Clearly, the original form of f is not its partial-fraction decomposition. In Table 2, the rational function converted to partial fractions has polynomials of the same degree for both numerator and denominator. If the degree of the numerator is greater than, or equal to, the degree of the denominator, a long division is required. The partial-fraction decomposition of the remainder divided by the divisor is then obtained. By long division, obtain the following, and apply the decomposition to the reduced rational function (the rightmost fraction), which is the remainder divided by the divisor.
Table 3 shows the partial-fraction decomposition of the reduced rational function. Control-drag the reduced rational function. Context Menu: Conversions_Partial Fractions_x convert to partial fractions in x Table 3 Partial-fraction decomposition of the reduced rational function The Decomposition as an Identity Form and solve the identity for the decomposition Set the expression for f equal to the sum of the appropriate template fractions, and press the Enter key. Context Menu: Solve_As Identity (Complete the dialog as per figure.) (1) solve (identity) (2) Transfer parameter values to the identity Expression palette_evaluation template Reference the identity and the set of parameter values by equation labels and press the Enter key.
(3) A Decomposition Task Template The task template in Table 4 applies the Maple command that renders a partial-fraction decomposition. The Tab key will advance the cursor and select the field in which the function to be decomposed is entered. Simply press the Enter key for each cell on the right of the table. Tools_Tasks_Browse: Algebra_Partial Fractions_Decomposition Enter a rational function: > Partial Fraction Decomposition (4) Enter the variable of the decomposition: > x (5) Convert to partial-fraction form: > (6) Table 4 Decomposition by task template
Stepwise Partial-Fractions Task Template Table 5 contains the Stepwise Partial Fraction Decomposition task template applied to the function f. The rational function to be decomposed can be referenced by its name, f. The long-division has to be anticipated in the box where the template fractions are entered. Note the check that the template fractions are correct. It would be senseless to execute the remaining steps of the decomposition if the templates themselves were not correct. The task template illustrates one of the many algebraic techniques that can be used to obtain the partial-fraction decomposition. For the student who knows no other, it is a viable starting point for mastering the requisite algebraic skills. Tools_Tasks_Browse: Algebra_Partial Fractions Stepwise Stepwise Partial Fraction Decomposition
Table 5 Decomposition by the stepwise task template Stepwise Partial Fractions by First Principles Table 6 implements from first principles the algorithm used by the task template in Table 5. Equation (7) equates the function and the template for the decomposition. The Context Menu is used to move all terms to the left and change the equation to an expression. This results in Equation (9), which could be simplified by adding fractions. However, it is the numerator of that fraction that must be identically zero; Maple can extract that numerator without explicitly obtaining the simplified fraction. This is the result in Equation (10), which has to have the coefficients of like powers of x set equal to zero. Hence, the Context Menu is first used to collect like terms, then to extract a sequence of coefficients of powers of x. This results in Equation (12), which is subjected to the Solve option in the Context Menu. Maple's solving commands all assume that an expression is equated to zero, so there is no need to explicitly form equations at this stage. The solution consists of a set of equations defining the values of the parameters. These values are inserted into the contents of Equation (7) via the Expression palette's Evaluation template.
Enter the appropriate identity and press the Enter key. Context Menu: Move to Left Left-hand Side Numerator Collect_ Coefficients_ Solve_Solve Expression palette: Evaluation template_transfer the values of the coefficients to the identity (7) move to left (8) left hand side (9) numerator collect w.r.t. x coefficients in x solve (10) (11) (12) (13) (14) Table 6 Stepwise implementation of the partial-fraction decomposition from first principles Pedagogical Note on Forming the Template Fractions Thinking back to when I first learned the partial-fraction "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 the "messier" affine numerator. Somewhere between learning this as a student and articulating it for students, I formulated the "rules"
simple - simple messy - 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. Implementing Long Division in Maple There are no syntax-free tools for implementing the long division of polynomials in Maple. The quo command must be used. Table 7 applies this command to. Table 7 The quo command for long division of polynomials 1 The arguments to quo are the numerator and denominator of the rational function, the variable (since the polynomials could have parameters as coefficients), and a placeholder for the remainder. In Table 7, the placeholder was taken as r, and the quotes around this name are essential. (If the name already has a value, an error will result. The quotes allow any previous value to be overwritten.) Decomposition to Linear Factors Consider the following partial-fraction decomposition. The quadratic factors over the real numbers since its zeros are. Hence, it should be possible to further decompose the rational function to fractions with just linear denominators. Table 8 illustrates how to obtain the partial-fraction decomposition in terms of fractions with linear denominators. (The option "fully reduced" in the Context Menu invokes the fullparfrac option in the underlying Maple command.) Control-drag the rational function and press the Enter key. Context Menu: Conversions_Partial Fractions (fully reduced)_x Context Menu: Conversions_To Radical
convert to reduced partial fractions in x radical form Table 8 A partial-fraction decomposition fully reduced The zeros of the denominator of the rational function are the complex numbers. This rational function can be decomposed to fractions with linear denominators using the same tools that were used in Table 8. This is done in Table 9. Control-drag the rational function; press the Enter key. Context Menu: Conversions_Partial Fractions (fully reduced)_x Context Menu: Conversions_To Radical convert to reduced partial fractions in x radical form Table 9 A complete partial-fraction decomposition over the complex field
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