The PartialFractionSteps command accepts a polynomial and displays the steps required to perform a partial fraction decomposition.

If no variable is given, then a variable is chosen by calling indets.  When the expression is multivariate, the first indeterminate is used.

If expr is a string, then it is parsed into an expression using InertForm:-Parse so that no automatic simplifications are applied, and thus no steps are missed.

The implicitmultiply option is only relevant when expr is a string.  This option is passed directly on to the InertForm:-Parse command and will cause things like 2x to be interpreted as 2*x, but also, xyz to be interpreted as x*y*z.

The output and displaystyle options are described in Student:-Basics:-OutputStepsRecord. The return value is controlled by the output option.

This function is part of the Student:-Basics package.

$\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right)\:$

$\mathrm{PartialFractionSteps}\left(-\frac{4x}{\left(x+3\right)\left(x+2\right)}\right)$

$\begin{array}{lll}& & -\frac{4x}{\left(x+3\right)\left(x+2\right)}\\ \text{•}& & \text{First, factor the numerator and denominator. If we can simplify by dividing terms, we do so.}\\ & & -\frac{4x}{\left(x+3\right)\left(x+2\right)}\\ \text{•}& & \text{Use the factors to write an equation for the partial fraction decomposition}\\ & & -\frac{4x}{\left(x+3\right)\left(x+2\right)}=\frac{A}{x+2}+\frac{B}{x+3}\\ \text{•}& & \text{Multiply by the LHS's denominator.}\\ & & -4x=\left(\frac{A}{x+2}+\frac{B}{x+3}\right)\left(x+3\right)\left(x+2\right)\\ \text{•}& & \text{Expand everything out.}\\ & & -4x=Ax+Bx+3A+2B\\ \text{•}& & \text{Equate the coefficients to create equations}\\ & & \left[0=3A+2B\,\mathrm{-4}=A+B\right]\\ \text{•}& & \text{Solve the equations.}\\ & & \left[A=8\,B=\mathrm{-12}\right]\\ \text{•}& & \text{Use these values in the partial fraction decomposition and simplify. The final partial fraction decomposition is}\\ & & \frac{8}{x+2}-\frac{12}{x+3}\end{array}$

The Student[Basics][PartialFractionSteps] command was introduced in Maple 2024.

For more information on Maple 2024 changes, see Updates in Maple 2024.