Solution by Context Panel
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Control-drag the rational function.
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Context Panel: Conversions≻Partial Fractions≻
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When working interactively, this solution via the Context Panel is probably the most efficient. Of course, there is the command-based conversion to partial fractions, as illustrated in Table 6.4.5(a).
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Assign the name to the given rational function.
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Apply the convert command, with the option parfrac, and the independent variable .
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Table 6.4.5(a) Use of the convert command to obtain a partial-fraction decomposition
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The stepwise partial-fractions task template in Table 6.4.5(b) provides an interactive stepwise solution.
Solution by Task Template
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Tools≻Tasks≻Browse: Algebra≻Partial Fractions≻Stepwise
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Stepwise Partial Fraction Decomposition
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Write rational function here
Write the partial-fraction decomposition template in this box
≡
*
To determine the constants, multiply both sides of the identity (*) by the denominator of the fraction on the left.
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Table 6.4.5(b) Task template for a stepwise interactive partial-fraction decomposition
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Table 6.4.5(c) contains an interactive solution from first principles. A similar calculation in the form of a recorded demo can be found here.
Interactive solution from first principles
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Control-drag the given rational function.
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Context Panel: Assign to a Name≻
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Enter the decomposition template.
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Context Panel: Assign to a Name≻
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Write and press the Enter key.
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Context Panel: Simplify≻Simplify
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Context Panel: Numerator
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Context Panel: Collect≻
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Context Panel: Coefficients≻
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Context Panel: Solve≻Solve
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Context Panel: Assign to a Name≻
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Expression palette: Evaluation template
Evaluate the template with the parameters in
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Table 6.4.5(c) Interactive solution from first principles
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Table 6.4.5(d) contains a coded solution from first principles.
Coded solution from first principles
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Write the basic identity for the partial-fraction decomposition.
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Multiply through by the denominator of the rational function.
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Form the equation determined by identifying the coefficients of powers of .
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Form the equation determined by identifying the coefficients of powers of .
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Form the equation determined by identifying the coefficients of powers of .
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Form the equation determined by identifying the coefficients of powers of .
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Form the equation determined by identifying the coefficients of powers of .
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Solve the four equations in .
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Evaluate the basic partial-fractions identity with the values of the coefficients determined by the solve command.
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Table 6.4.5(d) Coded solution from first principles
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Essentially, the partial-fraction decomposition seeks to determine the parameters , that make equation (in Table 6.4.5(d)) an identity in . This can be done immediately with the following modification of Maple's solve command.