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Student[Basics]

 ExpandSteps
 generates core steps in expanding polynomial products

 Calling Sequence Student[Basics][ExpandSteps]( expr ) Student[Basics][ExpandSteps]( expr, implicitmultiply = true )

Parameters

 expr - string or expression implicitmultiply - truefalse (optional)

Description

 • The ExpandSteps command accepts a product of polynomials and displays the steps required to expand the expression.
 • 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.
 • A step may show up where the expression is not obviously different from the previous step.  This can happen when the underlying data structure is transformed during the step, and it is not obvious that the resulting structure is the same as the original, but just expressed differently.  This becomes more apparent when looking at the inert-form of the raw data.
 • The return value is a module that displays annotated steps by default.  This module also has callable methods and data members: data, numsteps, step, and toMathML.

data: a numsteps x 2 array where column 1 is the inert-form expression, and column 2 is the annotation.  R:-data[1,1] is the original expression in inert-form.

numsteps: the number of steps in the solution, including the original expression.

step(i): a method for displaying individual steps.  Calling R:-step(i) will display the ith typeset expression and annotation.  Step 1 is the original expression.

toMathML(): a method for converting the sequence of steps and annotations into mathml.  The toMathML command optionally takes one or two arguments: (1) a filename, indicating the mathml should be written to the specified file, and (2) the option htmlheader=true, which will also cause html tags to be written along with the mathml, thus generating a complete .html page that can be loaded in a browser.

Package Usage

 • This function is part of the Student[Basics] package, so it can be used in the short form ExpandSteps(..) only after executing the command with(Student[Basics]). However, it can always be accessed through the long form of the command by using Student[Basics][ExpandSteps](..).

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{Basics}]\right):$
 > $\mathrm{ExpandSteps}\left(\frac{{a}^{2}-1}{\frac{a}{3}+\frac{1}{3}}\right)$
 $\begin{array}{c}\frac{{{a}}^{{2}}{-}{1}}{\frac{{a}}{{3}}{+}\frac{{1}}{{3}}}\\ {=}\frac{\left({a}{+}{1}\right){}\left({a}{-}{1}\right)}{\left({a}{+}{1}\right){}\frac{{1}}{{3}}}& \left({\mathrm{factor}}\right)\\ {=}\frac{{a}{-}{1}}{\frac{{1}}{{3}}}& \left({\mathrm{divide}}\right)\\ {=}\left({a}{-}{1}\right){}\frac{{3}}{{1}}& \left({\mathrm{rewrite division as multiplication by reciprocal}}\right)\\ {=}{3}{}{a}{-}{3}& \left({\mathrm{multiply fraction and reduce by gcd}}\right)\end{array}$ (1)

Note that the result is a module with callable methods

 > $\mathrm{ex}≔\mathrm{ExpandSteps}\left(3a\left(4a-y+42\right)\right)$
 ${\mathrm{ex}}{≔}\begin{array}{c}{3}{}{a}{}\left({4}{}{a}{+}\left({-1}\right){}{y}{+}{42}\right)\\ {=}{3}{}{a}{·}{4}{}{a}{+}{3}{}{a}{}\left({-}{y}\right){+}{3}{}{a}{·}{42}& \left({\mathrm{distributive multiply}}\right)\\ {=}{12}{}{a}{}{a}{+}{3}{}{a}{}\left({-}{y}\right){+}{3}{}{a}{·}{42}& \left({\mathrm{multiply constants}}\right)\\ {=}{12}{}{{a}}^{{2}}{+}{3}{}{a}{}\left({-}{y}\right){+}{3}{}{a}{·}{42}& \left({\mathrm{multiply terms to exponential form}}\right)\\ {=}{12}{}{{a}}^{{2}}{+}\left({-3}\right){}{a}{}{y}{+}{3}{}{a}{·}{42}& \left({\mathrm{multiply constants}}\right)\\ {=}{12}{}{{a}}^{{2}}{-}{3}{}{a}{}{y}{+}{126}{}{a}& \left({\mathrm{multiply constants}}\right)\end{array}$ (2)
 > $\mathrm{ex}:-\mathrm{numsteps}$
 ${6}$ (3)
 > $\mathrm{ex}:-\mathrm{step}\left(2\right)$
 $\begin{array}{cc}{3}{}{a}{·}{4}{}{a}{+}{3}{}{a}{}\left({-}{y}\right){+}{3}{}{a}{·}{42}& \left({\mathrm{distributive multiply}}\right)\end{array}$ (4)

The steps can be converted to MathML

 > $\mathrm{ex}:-\mathrm{toMathML}\left(\right)$
 $\left[\begin{array}{cccccc}{"3 a 4 a+-1 y+42\left[/itex\right]"}& {"=3 a 4 a+3 a -y+3 a 42\left( distributive multiply \right)\left[/itex\right]"}& {"=12 a a+3 a -y+3 a 42\left( multiply constants \right)\left[/itex\right]"}& {"=12 ^a2+3 a -y+3 a 42\left( multiply terms to exponential form \right)\left[/itex\right]"}& {"=12 a2+-3 a y+3 a 42\left( multiply constants \right)\left[/itex\right]"}& {"=12 a2+-3 a y+126 a\left( multiply constants \right)\left[/itex\right]"}\end{array}\right]$ (5)

The input can be a string, which prevents automatic simplification

 > $\mathrm{ExpandSteps}\left("\left(1+1\right)*\left(3-1\right)"\right)$
 $\begin{array}{c}\left({1}{+}{1}\right){}\left({3}{-}{1}\right)\\ {=}{2}{}\left({3}{-}{1}\right)& \left({\mathrm{add terms}}\right)\\ {=}{2}{·}{2}& \left({\mathrm{add terms}}\right)\\ {=}{4}& \left({\mathrm{multiply constants}}\right)\end{array}$ (6)

The implicitmultiply option allows short-hand for string input.

 > $\mathrm{ExpandSteps}\left("2.1yx^2/\left(4.3xy\right)",'\mathrm{implicitmultiply}'\right)$
 $\begin{array}{c}\frac{{2.1}{}{y}{}{{x}}^{{2}}}{{4.3}{}{x}{}{y}}\\ {=}\frac{{2.1}{}{{x}}^{{2}}}{{4.3}{}{x}}& \left({\mathrm{divide out common terms}}\right)\\ {=}\frac{{2.100000000}{}{x}}{{4.3}}& \left({\mathrm{divide out common terms}}\right)\\ {=}{0.4883720930}{}{x}& \left({\mathrm{divide constants}}\right)\end{array}$ (7)

Compatibility

 • The Student[Basics][ExpandSteps] command was introduced in Maple 18.