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Classroom Tips and Techniques:The Partial-Fraction DecompositionRobert J. LopezEmeritus Professor of Mathematics and Maple FellowMaplesoft

JSFH 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. JSFHJSFH

JSFH Table 1 examines how Maple's Context Menu can be invoked to sum two algebraic fractions. JSFH Define the fractions to be added Write the sum of two fractions. Context Menu: Assign to a Name\342\211\273f 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

= LCYqJiwmKiYiIiMiIiJJInhHNiJGJ0YnRidGJ0YnLCYqJClGKEYmRidGJ0YnRichIiJGJyomLCZGKEYnRidGLUYnLCZGKEYnIiIkRidGLUYn

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMWFzc2lnbn50b35hfm5hbWVGJ0YvRjJGNUZF f := : 'f';

SSJmRzYi

Simplify Write the name f. Context Menu: Evaluate and Display Inline Context Menu: Simplify\342\211\273Simplify LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

= LCYqJiwmKiYiIiMiIiJJInhHNiJGJ0YnRidGJ0YnLCYqJClGKEYmRidGJ0YnRichIiJGJyomLCZGKEYnRidGLUYnLCZGKEYnIiIkRidGLUYn

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RIj1GJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSZtdGV4dEdGJDYmUSt+c2ltcGxpZnl+RidGL0YyRjVGRQ== simplify( );

KigsKiokKUkieEc2IiIiJCIiIkYpKiQpRiYiIiNGKUYpKiYiIilGKUYmRilGKUYsRilGKSwmRipGKUYpRikhIiIsJkYmRilGKEYpRjA=

Normalize Write the name f. Context Menu: Evaluate and Display Inline Context Menu: Normal LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

= LCYqJiwmKiYiIiMiIiJJInhHNiJGJ0YnRidGJ0YnLCYqJClGKEYmRidGJ0YnRichIiJGJyomLCZGKEYnRidGLUYnLCZGKEYnIiIkRidGLUYn

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZGN0YvRjJGNUZF normal( );

KigsKiokKUkieEc2IiIiJCIiIkYpKiQpRiYiIiNGKUYpKiYiIilGKUYmRilGKUYsRilGKSwmRipGKUYpRikhIiIsJkYmRilGKEYpRjA=

Normalize but expand the denominator Write the name f. Context Menu: Evaluate and Display Inline Context Menu: Simplify\342\211\273Normalize Expanded LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

= LCYqJiwmKiYiIiMiIiJJInhHNiJGJ0YnRidGJ0YnLCYqJClGKEYmRidGJ0YnRichIiJGJyomLCZGKEYnRidGLUYnLCZGKEYnIiIkRidGLUYn

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRM25vcm1hbGl6ZX5leHBhbmRlZEYnRi9GMkY1RkU= normal( , ':-expanded' );

KiYsKiokKUkieEc2IiIiJCIiIkYpKiQpRiYiIiNGKUYpKiYiIilGKUYmRilGKUYsRilGKSwqRiRGKSomRihGKUYrRilGKUYmRilGKEYpISIi

Table 1 Maple's syntax-free tools for adding two algebraic fractions JSFH

JSFH 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\342\211\273Partial Fractions\342\211\273x 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

= KiYsKiokKUkieEc2IiIiJCIiIkYpKiQpRiYiIiNGKUYpKiYiIilGKUYmRilGKUYsRilGKSwqRiRGKSomRihGKUYrRilGKUYmRilGKEYpISIi

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RIj1GJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSZtdGV4dEdGJDYmUUR+Y29udmVydH50b35wYXJ0aWFsfmZyYWN0aW9uc35pbn54fkYnRi9GMkY1RkU= convert( , 'parfrac', x );

LCgiIiJGIyomIiIlRiMsJkkieEc2IkYjIiIkRiMhIiJGKiomLCYqJiIiI0YjRidGI0YjRiNGI0YjLCYqJClGJ0YuRiNGI0YjRiNGKkYj

Table 2 Partial-fraction decomposition via the Context Menu JSFH 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. JSFH 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. 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 JSFH Table 3 shows the partial-fraction decomposition of the reduced rational function. Control-drag the reduced rational function. Context Menu: Conversions\342\211\273Partial Fractions\342\211\273x 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

= LCQqKCwoKiYiIiMiIiIpSSJ4RzYiRiZGJ0YnKiYiIihGJ0YpRichIiJGJ0YnRicsJkYpRiciIiRGJ0YtLCYqJEYoRidGJ0YnRidGLUYt

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RIj1GJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSZtdGV4dEdGJDYmUUR+Y29udmVydH50b35wYXJ0aWFsfmZyYWN0aW9uc35pbn54fkYnRi9GMkY1RkU= convert( , 'parfrac', x );

LCYqJiIiJSIiIiwmSSJ4RzYiRiUiIiRGJSEiIkYqKiYsJiomIiIjRiVGJ0YlRiVGJUYlRiUsJiokKUYnRi5GJUYlRiVGJUYqRiU=

Table 3 Partial-fraction decomposition of the reduced rational function JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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\342\211\273As Identity (Complete the dialog as per figure.) 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

LywmKiYsJiomIiIjIiIiSSJ4RzYiRihGKEYoRihGKCwmKiQpRilGJ0YoRihGKEYoISIiRigqJiwmRilGKEYoRi5GKCwmRilGKCIiJEYoRi5GKCwoRihGKComSSJhR0YqRihGMUYuRigqJiwmKiZJImJHRipGKEYpRihGKEkiY0dGKkYoRihGK0YuRig=

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMXNvbHZlfihpZGVudGl0eSlGJ0YvRjJGNUZF solve( identity( , x ), {a, b, c} );

PCUvSSJhRzYiISIlL0kiYkdGJSIiIy9JImNHRiUiIiI=

Transfer parameter values to the identity Expression palette\342\211\273Evaluation template Reference the identity and the set of parameter values by equation labels and press the Enter key. 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

LywmKiYsJiomIiIjIiIiSSJ4RzYiRihGKEYoRihGKCwmKiQpRilGJ0YoRihGKEYoISIiRigqJiwmRilGKEYoRi5GKCwmRilGKCIiJEYoRi5GKCwoRihGKComIiIlRihGMUYuRi5GJEYo

JSFH

JSFH 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. JSFH Tools\342\211\273Tasks\342\211\273Browse: Algebra\342\211\273Partial Fractions\342\211\273Decomposition Partial Fraction Decomposition Enter a rational function: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSShtYWN0aW9uR0YkNiQtRiM2Ji1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxwbGFjZWhvbGRlckdGOS9GO1Enbm9ybWFsRicvJSthY3Rpb250eXBlR1E7bWFwbGVzb2Z0OnRhc2stcGxhY2Vob2xkZXJGJ0YrRj1GQg==

LCYqJiwmKiYiIiMiIiJJInhHNiJGJ0YnRidGJ0YnLCYqJClGKEYmRidGJ0YnRichIiJGJyomLCZGKEYnRidGLUYnLCZGKEYnIiIkRidGLUYn

Enter the variable of the decomposition: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2J0YrLUkobWFjdGlvbkdGJDYkLUYjNiYtRiw2J1EieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRK1s5NSwwLDIxN11GJy8lMGZvbnRfc3R5bGVfbmFtZUdROE1hcGxlfklucHV0flBsYWNlaG9sZGVyRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxwbGFjZWhvbGRlckdGOy9GQ1Enbm9ybWFsRicvJSthY3Rpb250eXBlR1E7bWFwbGVzb2Z0OnRhc2stcGxhY2Vob2xkZXJGJ0YrRkVGSkYrRkVGSg==

SSJ4RzYi

Convert to partial-fraction form: 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

LCgiIiJGIyomIiIlRiMsJkkieEc2IkYjIiIkRiMhIiJGKiomLCYqJiIiI0YjRidGI0YjRiNGI0YjLCYqJClGJ0YuRiNGI0YjRiNGKkYj

Table 4 Decomposition by task template JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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\342\211\273Tasks\342\211\273Browse: Algebra\342\211\273Partial Fractions Stepwise Stepwise Partial Fraction Decomposition use DocumentTools in Do(%MathContainer12 = NULL); Do(%MathContainer13 = NULL); Do(%MathContainer14 = NULL); Do(%MathContainer15 = NULL); Do(%MathContainer16 = NULL); Do(%MathContainer17 = NULL); Do(%MathContainer18 = NULL); Do(%MathContainer19 = NULL); Do(%MathContainer20 = NULL); Do(%MathContainer21 = NULL); Do(%MathContainer22 = NULL); Do(%MathContainer23 = NULL); Do(%Button18(enabled)=false); Do(%Button20(enabled)=false); Do(%Button22(enabled)=false); Do(%Button24(enabled)=false); Do(%Button28(enabled)=false); Do(%Button26(enabled)=false); Do(%MathContainer15(pixelWidth)=100); Do(%MathContainer20(pixelHeight)=60); end use; Write rational function here 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LSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIkUyRGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiQ1I0YqLSUpQ0hJTERSRU5HNiI= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEnJmRhcnI7RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZGRi9GMg== use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. module() local f,ans; Do(f = %MathContainer12); if nops([f])=0 then return;end if; ans := numer(f)/factor(denom(f)); Do(%MathContainer13 = ans); Do(%Button18(enabled)=true); end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(%MathContainer12 = NULL); Do(%MathContainer13 = NULL); end use; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRkZGL0Yy Write the partial-fraction decomposition template in this box TUZOV3RLVWI8b2I8Uj1NRExDZE5iQlI/Q3I6Um5nUkJDQVtDTlNET0U8a0JAa0I8S0I8W0NGW0NKO0RKW0ROW3RZeXVZa0VYa3VbSXZbSXdjSXdlWXdleXhxeXhrSXhraXl3aXl1WXl1eXl5eWlYeUk9Zz5LVUxDVEpjRFhveFJzeW07ZjtrZGlfSUJrQ115SUo/Q3FLZFxcd2NaS2JDY1VNQUJFP2g6SVVxU3lSa1lrR0hqW1hcXFtSTnNWOz9TcEllQmNVZmVCWktIPEN1Vm9UWmtWaD1SUEN2ajtWWj1kSj1SUENWZF1WWk11PE9XdXdoPFNpO0toZ0FSPU1DWFlIWk9jaltWa09mdUdFXkhac0duSU5odWlmWz9IPFpDZ2JIX2JoUGJaT0o6Ojo6Ojo0OlwiXHtcfQ==LSUrQU5OT1RBVElPTkc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIkUyRGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiQ1I0YqLSUpQ0hJTERSRU5HNiI= \342\211\241 * LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEnJmRhcnI7RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZGRi9GMg== use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. module() local f,stuAns,partials,cnts,cntsStuAns,isZero,ans; Do(f = %MathContainer12); if nops([f])=0 then return;end if; Do(stuAns=%MathContainer14); if stuAns=NULL then Do(%MathContainer15(pixelWidth)=450); Do(%MathContainer15 = MathML[Export]("Please provide the decomposition template fractions.")); return; end if; partials:=convert(f,parfrac,x); cnts:=solve(identity(stuAns=partials,x)); if nops([cnts])=0 then Do(%MathContainer15(pixelWidth)=650); Do(%MathContainer15 = MathML[Export]("Maple claims the template fractions are not correct. Please check your work.")); return;end if; cntsStuAns:=eval(stuAns,cnts); isZero:=simplify(partials-cntsStuAns); if isZero=0 then Do(%MathContainer15(pixelWidth)=100); Do(%MathContainer15 = MathML[Export]("Correct!")); Do(%Button20(enabled)=true); else Do(%MathContainer15 = MathML[Export]("Maple claims the template fractions are not correct. Please check your work."));end if; end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(%MathContainer14 = NULL); Do(%MathContainer15 = NULL); Do(%MathContainer15(pixelWidth)=100); end use; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRkZGL0Yy LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRkZGL0Yy To determine the constants, multiply both sides of the identity (*) by the denominator of the fraction on the left. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEnJmRhcnI7RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZGRi9GMg== use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. module() local f,stuAns,partials,cnts,cntsStuAns,rightNumer,isZero,ans; Do(f = %MathContainer12); if nops([f])=0 then return;end if; Do(stuAns=%MathContainer14); partials:=convert(f,parfrac,x); cnts:=solve(identity(stuAns=partials,x)); cntsStuAns:=eval(stuAns,cnts); isZero:=simplify(partials-cntsStuAns); if isZero=0 then ans:='correct!' elif isZero!=0 then ans:='incorrect' fi; rightNumer:=simplify(stuAns*factor(denom(f))); leftNumer:=numer(f); Do(%MathContainer16 = leftNumer); Do(%MathContainer17 = rightNumer); Do(%Button22(enabled)=true); end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(%MathContainer16 = NULL); Do(%MathContainer17 = NULL); end use; LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YuLyUqc2VwYXJhdG9yR0YuLyUpc3RyZXRjaHlHRi4vJSpzeW1tZXRyaWNHRi4vJShsYXJnZW9wR0YuLyUubW92YWJsZWxpbWl0c0dGLi8lJ2FjY2VudEdGLi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGRUYvLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRjI= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEnJmRhcnI7RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZGRi9GMg== use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. module() local f,stuAns,partials,cnts,cntsStuAns,rightNumer,isZero,collectAll,ans; Do(f = %MathContainer12); if nops([f])=0 then return;end if; Do(stuAns=%MathContainer14); partials:=convert(f,parfrac,x); cnts:=solve(identity(stuAns=partials,x)); cntsStuAns:=eval(stuAns,cnts); isZero:=simplify(partials-cntsStuAns); if isZero=0 then ans:='correct!' elif isZero!=0 then ans:='incorrect' fi; rightNumer:=simplify(stuAns*factor(denom(f))); leftNumer:=numer(f); collectAll:=collect(rightNumer-leftNumer,x); Do(%MathContainer18 = 0); Do(%MathContainer19 = collectAll); Do(%Button24(enabled)=true); end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(%MathContainer18 = NULL); Do(%MathContainer19 = NULL); end use; LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YuLyUqc2VwYXJhdG9yR0YuLyUpc3RyZXRjaHlHRi4vJSpzeW1tZXRyaWNHRi4vJShsYXJnZW9wR0YuLyUubW92YWJsZWxpbWl0c0dGLi8lJ2FjY2VudEdGLi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEiPUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGRUYvLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRjI= JSFH LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YuLyUqc2VwYXJhdG9yR0YuLyUpc3RyZXRjaHlHRi4vJSpzeW1tZXRyaWNHRi4vJShsYXJnZW9wR0YuLyUubW92YWJsZWxpbWl0c0dGLi8lJ2FjY2VudEdGLi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkI= use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. module() local f,stuAns,partials,cnts,cntsStuAns,rightNumer,isZero,collectAll,ans,ans2,d,k,L,temp; Do(f = %MathContainer12); if nops([f])=0 then return;end if; Do(stuAns=%MathContainer14); partials:=convert(f,parfrac,x); cnts:=solve(identity(stuAns=partials,x)); cntsStuAns:=eval(stuAns,cnts); isZero:=simplify(partials-cntsStuAns); if isZero=0 then ans:='correct!' elif isZero!=0 then ans:='incorrect' fi; rightNumer := simplify(stuAns*factor(denom(f))); leftNumer := numer(f); collectAll := collect(rightNumer-leftNumer,x); d := degree(collectAll); L := NULL; for k from 0 to d do temp := coeff(collectAll,x,k); if temp<>0 then L:=L,temp;end if; end do; ans2 := L; N:=nops([ans2]); Z:=LinearAlgebra:-ZeroVector(N); v:=Vector([ans2])=Z; Do(%MathContainer20(pixelHeight)=N*40); Do(%MathContainer20=MathML[Export](v)): Do(%Button26(enabled)=true); end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(%MathContainer20 = NULL); Do(%MathContainer21 = NULL); end use; LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YuLyUqc2VwYXJhdG9yR0YuLyUpc3RyZXRjaHlHRi4vJSpzeW1tZXRyaWNHRi4vJShsYXJnZW9wR0YuLyUubW92YWJsZWxpbWl0c0dGLi8lJ2FjY2VudEdGLi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkI=JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHRkZGL0Yy JSFHuse DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. module() local f, stuAns, cnts; Do(f = %MathContainer12); if nops([f])=0 then return;end if; Do(stuAns=%MathContainer14); partials:=convert(f,parfrac,x); cnts := solve(identity(stuAns=partials,x)); Do(%MathContainer21 = cnts); Do(%Button28(enabled)=true); end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(%MathContainer21 = NULL); end use; \342\206\223 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFH LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YuLyUqc2VwYXJhdG9yR0YuLyUpc3RyZXRjaHlHRi4vJSpzeW1tZXRyaWNHRi4vJShsYXJnZW9wR0YuLyUubW92YWJsZWxpbWl0c0dGLi8lJ2FjY2VudEdGLi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. module() local f,stuAns,partials,cnts,cntsStuAns,rightNumer,isZero,collectAll,ans,ans2,ans3; Do(f = %MathContainer12); if nops([f])=0 then return;end if; Do(stuAns=%MathContainer14); partials:=convert(f,parfrac,x); cnts:=solve(identity(stuAns=partials,x)); cntsStuAns:=eval(stuAns,cnts); isZero:=simplify(partials-cntsStuAns); if isZero=0 then ans:='correct!' elif isZero!=0 then ans:='incorrect' fi; rightNumer:=simplify(stuAns*factor(denom(f))); leftNumer:=numer(f); collectAll:=collect(rightNumer-leftNumer,x); ans2:=coeffs(rightNumer-leftNumer,x); ans3:=partials; Do(%MathContainer23 = ans3); Do(%MathContainer22 = f); Do(%MathContainer23 = ans3); end module; end use; use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. Do(%MathContainer22 = NULL); Do(%MathContainer23 = NULL); end use; LUkjbW9HNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2LlEsJkRvd25BcnJvdztGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0YuLyUqc2VwYXJhdG9yR0YuLyUpc3RyZXRjaHlHRi4vJSpzeW1tZXRyaWNHRi4vJShsYXJnZW9wR0YuLyUubW92YWJsZWxpbWl0c0dGLi8lJ2FjY2VudEdGLi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEieEYnRi9GMkY1RjIvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RIj1GJ0YyRkAvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZWRjJGQA==JSFH= JSFH Table 5 Decomposition by the stepwise task template LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JlEiYkYnRi9GMkY1RjgtRiw2JlEiY0YnRi9GMkY1RjJGPA==. 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\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn Coefficients\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn Solve\342\211\273Solve Expression palette: Evaluation template\342\211\273Transfer the values of the coefficients to the identity 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

LywmKiYsJiomIiIjIiIiSSJ4RzYiRihGKEYoRihGKCwmKiQpRilGJ0YoRihGKEYoISIiRigqJiwmRilGKEYoRi5GKCwmRilGKCIiJEYoRi5GKCwoRihGKComSSJhR0YqRihGMUYuRigqJiwmKiZJImJHRipGKEYpRihGKEkiY0dGKkYoRihGK0YuRig=

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRLW1vdmV+dG9+bGVmdEYnRi9GMkY1RkU= lhs() - rhs() = 0;

LywsKiYsJiomIiIjIiIiSSJ4RzYiRihGKEYoRihGKCwmKiQpRilGJ0YoRihGKEYoISIiRigqJiwmRilGKEYoRi5GKCwmRilGKCIiJEYoRi5GKEYoRi4qJkkiYUdGKkYoRjFGLkYuKiYsJiomSSJiR0YqRihGKUYoRihJImNHRipGKEYoRitGLkYuIiIh

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRL2xlZnR+aGFuZH5zaWRlRidGL0YyRjVGRQ== lhs( );

LCwqJiwmKiYiIiMiIiJJInhHNiJGJ0YnRidGJ0YnLCYqJClGKEYmRidGJ0YnRichIiJGJyomLCZGKEYnRidGLUYnLCZGKEYnIiIkRidGLUYnRidGLSomSSJhR0YpRidGMEYtRi0qJiwmKiZJImJHRilGJ0YoRidGJ0kiY0dGKUYnRidGKkYtRi0=

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRKm51bWVyYXRvckYnRi9GMkY1RkU= numer( );

LDQqJkkiYUc2IiIiIilJInhHRiUiIiNGJiEiIiomSSJiR0YlRiZGJ0YmRioqKCIiJEYmRixGJkYoRiZGKiomSSJjR0YlRiZGKEYmRioqJkYpRiZGJ0YmRipGJEYqKiZGLkYmRjBGJkYqKiYiIihGJkYoRiZGJkYmRio=

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RIj1GJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSZtdGV4dEdGJDYmUTN+Y29sbGVjdH53LnIudC5+eH5GJ0YvRjJGNUZF collect( , x );

LCwqJiwoSSJhRzYiISIiSSJiR0YmRiciIiNGJyIiIilJInhHRiZGKUYqRioqJiwoKiYiIiRGKkYoRipGJ0kiY0dGJkYnIiIoRipGKkYsRipGKkYlRicqJkYwRipGMUYqRidGKkYn

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRMmNvZWZmaWNpZW50c35pbn54RidGL0YyRjVGRQ== coeffs( collect( , x ), x );

NiUsKEkiYUc2IiEiIiomIiIkIiIiSSJjR0YlRilGJkYpRiYsKComRihGKUkiYkdGJUYpRiZGKkYmIiIoRiksKEYkRiZGLUYmIiIjRiY=

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RJyZyYXJyO0YnLyUrZXhlY3V0YWJsZUdRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKDJEfk1hdGhGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JJm10ZXh0R0YkNiZRJnNvbHZlRidGL0YyRjVGRQ== solve( { } );

PCUvSSJhRzYiISIlL0kiYkdGJSIiIy9JImNHRiUiIiI=

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

LywmKiYsJiomIiIjIiIiSSJ4RzYiRihGKEYoRihGKCwmKiQpRilGJ0YoRihGKEYoISIiRigqJiwmRilGKEYoRi5GKCwmRilGKCIiJEYoRi5GKCwoRihGKComIiIlRihGMUYuRi5GJEYo

Table 6 Stepwise implementation of the partial-fraction decomposition from first principles LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH 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" JSFH simple - simplemessy - messyWatch out for the repeats! JSFH 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.JSFH

JSFH 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 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. 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

= IiIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==

= LCgqJiIiIyIiIilJInhHNiJGJEYlISIiKiYiIihGJUYnRiVGJUYlRik=

Table 7 The quo command for long division of polynomials 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.) JSFH

JSFH Consider the following partial-fraction decomposition. 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 JSFH JSFH The quadratic 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 factors over the real numbers since its zeros are LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYlLUYjNiktSSNtb0dGJDYuUSomdW1pbnVzMDtGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkotSSNtbkdGJDYlUSIxRidGNEY3LUYxNi5RIn5GJ0Y0RjdGOkY8Rj5GQEZCRkRGRi9GSVEmMC4wZW1GJy9GTEZVLUYxNi5RJSZwbTtGJ0Y0RjdGOkY8Rj5GQEZCRkRGRkZIRkstSSZtc3FydEdGJDYjLUYjNiUtRk42JVEiNUYnRjRGN0Y0RjdGNEY3RjRGNy1GMTYuUSIvRidGNEY3RjpGPC9GP1EldHJ1ZUYnRkBGQkZERkYvRklRLDAuMTY2NjY2N2VtRicvRkxGYm8tRk42JVEiMkYnRjRGN0Y0Rjc=. 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.) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Control-drag the rational function and press the Enter key. Context Menu: Conversions\342\211\273Partial Fractions (fully reduced)\342\211\273x Context Menu: Conversions\342\211\273To Radical 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

KiosLComIiM2IiIiKUkieEc2IiIiJUYmRiYqJiIiKUYmKUYoIiIkRiZGJiomIiIqRiYpRigiIiNGJiEiIiomIiNCRiZGKEYmRjNGJUYmRiYsJkYoRiYiIiZGJkYzLCZGKEYmRi5GM0YzLCgqJEYxRiZGJkYoRiZGJkYzRjM=

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RIj1GJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSZtdGV4dEdGJDYmUUx+Y29udmVydH50b35yZWR1Y2VkfnBhcnRpYWx+ZnJhY3Rpb25zfmlufnh+RidGL0YyRjVGRQ== convert( , 'fullparfrac', x );

LCoiIzYiIiIqJiIjUUYkLCZJInhHNiJGJCIiJkYkISIiRisqJkYjRiQsJkYoRiQiIiRGK0YrRiQtSSRTdW1HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRik2JComLCYqJiMiIiVGKkYkSSdfYWxwaGFHRilGJEYrI0YuRipGJEYkLCZGKEYkRjpGK0YrL0Y6LUknUm9vdE9mR0YxNiMsKCokKUkjX1pHRjEiIiNGJEYkRkRGJEYkRitGJA==

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RIj1GJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSZtdGV4dEdGJDYmUS9+cmFkaWNhbH5mb3JtfkYnRi9GMkY1RkU= convert( , 'radical' );

LCwiIzYiIiIqJiIjUUYkLCZJInhHNiJGJCIiJkYkISIiRisqJkYjRiQsJkYoRiQiIiRGK0YrRiQqJiwmKiYjIiIjRipGJClGKiNGJEYzRiRGK0YkRiRGJCwoRihGJComRjVGJEY0RiRGK0Y1RiRGK0YkKiYsJkYkRiRGMUYkRiQsKEYoRiRGNUYkRjdGJEYrRiQ=

Table 8 A partial-fraction decomposition fully reduced LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The zeros of the denominator of the rational function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkmbWZyYWNHRiQ2KC1GIzYoLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiK0YnRjcvRjtRJ25vcm1hbEYnLyUmZmVuY2VHRjkvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGUy1JI21uR0YkNiVRIjFGJ0Y3RkFGNEY3RjotRiM2Ki1JJW1zdXBHRiQ2JUYwLUYjNiYtRlc2JVEiMkYnRjdGQUY0RjdGOi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGPUYwRj1GVkY0RjdGOi8lLmxpbmV0aGlja25lc3NHRlkvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZW8vJSliZXZlbGxlZEdGOUY3RkE= are the complex numbers LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkobWZlbmNlZEdGJDYlLUYjNistSSNtb0dGJDYuUSomdW1pbnVzMDtGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkotSSNtbkdGJDYlUSIxRidGNEY3LUYxNi5RIn5GJ0Y0RjdGOkY8Rj5GQEZCRkRGRi9GSVEmMC4wZW1GJy9GTEZVLUYxNi5RJSZwbTtGJ0Y0RjdGOkY8Rj5GQEZCRkRGRkZIRkstSSNtaUdGJDYmUSJpRicvJSdpdGFsaWNHUSV0cnVlRidGNC9GOFEnaXRhbGljRidGUS1JJm1zcXJ0R0YkNiMtRiM2JS1GTjYlUSIzRidGNEY3RjRGN0Y0RjdGNEY3LUYxNi5RIi9GJ0Y0RjdGOkY8L0Y/RmpuRkBGQkZERkYvRklRLDAuMTY2NjY2N2VtRicvRkxGam8tRk42JVEiMkYnRjRGN0Y0Rjc=. 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Control-drag the rational function; press the Enter key. Context Menu: Conversions\342\211\273Partial Fractions (fully reduced)\342\211\273x Context Menu: Conversions\342\211\273To Radical LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1GIzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIrRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVC1JI21uR0YkNiVRIjFGJ0Y3RkEtRiM2Jy1JJW1zdXBHRiQ2JUYwLUYjNiMtRlg2JVEiMkYnRjdGQS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGPUYwRj1GVy8lLmxpbmV0aGlja25lc3NHRlovJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZm8vJSliZXZlbGxlZEdGRS1GMTYjUSFGJw==

KiYsJkkieEc2IiIiIkYmRiZGJiwoKiQpRiQiIiNGJkYmRiRGJkYmRiYhIiI=

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RIj1GJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSZtdGV4dEdGJDYmUUx+Y29udmVydH50b35yZWR1Y2VkfnBhcnRpYWx+ZnJhY3Rpb25zfmlufnh+RidGL0YyRjVGRQ== convert( , 'fullparfrac', x );

LUkkU3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJiwmKiYjIiIiIiIkRi1JJ19hbHBoYUdGJ0YtISIiRixGLUYtLCZJInhHRidGLUYvRjBGMC9GLy1JJ1Jvb3RPZkdGJDYjLCgqJClJI19aR0YkIiIjRi1GLUY6Ri1GLUYt

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi9RIj1GJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSgyRH5NYXRoRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkktSSZtdGV4dEdGJDYmUS9+cmFkaWNhbH5mb3JtfkYnRi9GMkY1RkU= convert( , 'radical' );

LCYqJiwmIyIiIiIiI0YmKiYsJComI0YmIiInRiZeI0YmRiZGJkYmKSIiJEYlRiYhIiJGJiwoSSJ4RzYiRiZGJUYmKiYsJComRiVGJkYtRiZGJkYmRi5GJkYwRjBGJiomLCZGJUYmKiZGKkYmRi5GJkYmRiYsKEYyRiZGJUYmKiZGNkYmRi5GJkYmRjBGJg==

Table 9 A complete partial-fraction decomposition over the complex field LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH JSFH Legal Notice: \302\251 Maplesoft, a division of Waterloo Maple Inc. 2017. Maplesoft and Maple are trademarks of Waterloo Maple Inc. This application may contain errors and Maplesoft is not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact Maplesoft for permission if you wish to use this application in for-profit activities. JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH