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generates core steps in expanding polynomial products


Calling Sequence



Package Usage



Calling Sequence

Student[Basics][ExpandSteps]( expr )

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




string or expression



truefalse (optional)



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](..).




a21a3+13=a+1a1a+113factor=a113divide=a131rewrite division as multiplication by reciprocal=3a3multiply fraction and reduce by gcd


Note that the result is a module with callable methods


ex3a4a+−1y+42=3a·4a+3ay+3a·42distributive multiply=12aa+3ay+3a·42multiply constants=12a2+3ay+3a·42multiply terms to exponential form=12a2+−3ay+3a·42multiply constants=12a23ay+126amultiply constants






3a·4a+3ay+3a·42distributive multiply


The steps can be converted to MathML


<math xmlns=''><mstyle scriptminsize='8.0pt'><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mrow><mrow><mrow><mn>4</mn><mo> </mo><mi>a</mi></mrow><mo>+</mo><mrow><mn>-1</mn><mo> </mo><mi>y</mi></mrow></mrow><mo>+</mo><mn>42</mn></mrow></mrow></mstyle></math> <math xmlns=''><mstyle scriptminsize='8.0pt'><mtext>=</mtext><mspace width='5px'><mrow><mrow><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mrow><mn>4</mn><mo> </mo><mi>a</mi></mrow></mrow><mo>+</mo><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mfenced><mrow><mo>-</mo><mi>y</mi></mrow></mfenced></mrow></mrow><mo>+</mo><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mn>42</mn></mrow></mrow><mspace width='10px'><mtext color='blue'>( distributive multiply )</mtext></mstyle></math> <math xmlns=''><mstyle scriptminsize='8.0pt'><mtext>=</mtext><mspace width='5px'><mrow><mrow><mrow><mrow><mn>12</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mi>a</mi></mrow><mo>+</mo><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mfenced><mrow><mo>-</mo><mi>y</mi></mrow></mfenced></mrow></mrow><mo>+</mo><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mn>42</mn></mrow></mrow><mspace width='10px'><mtext color='blue'>( multiply constants )</mtext></mstyle></math> <math xmlns=''><mstyle scriptminsize='8.0pt'><mtext>=</mtext><mspace width='5px'><mrow><mrow><mrow><mn>12</mn><mo> </mo><mrow><mi>^</mi><mo></mo><mfenced><mi>a</mi><mn>2</mn></mfenced></mrow></mrow><mo>+</mo><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mfenced><mrow><mo>-</mo><mi>y</mi></mrow></mfenced></mrow></mrow><mo>+</mo><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mn>42</mn></mrow></mrow><mspace width='10px'><mtext color='blue'>( multiply terms to exponential form )</mtext></mstyle></math> <math xmlns=''><mstyle scriptminsize='8.0pt'><mtext>=</mtext><mspace width='5px'><mrow><mrow><mrow><mn>12</mn><mo> </mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mrow><mn>-3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mi>y</mi></mrow></mrow><mo>+</mo><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mn>42</mn></mrow></mrow><mspace width='10px'><mtext color='blue'>( multiply constants )</mtext></mstyle></math> <math xmlns=''><mstyle scriptminsize='8.0pt'><mtext>=</mtext><mspace width='5px'><mrow><mrow><mrow><mn>12</mn><mo> </mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo>+</mo><mfenced><mrow><mo>-</mo><mrow><mrow><mn>3</mn><mo> </mo><mi>a</mi></mrow><mo> </mo><mi>y</mi></mrow></mrow></mfenced></mrow><mo>+</mo><mrow><mn>126</mn><mo> </mo><mi>a</mi></mrow></mrow><mspace width='10px'><mtext color='blue'>( multiply constants )</mtext></mstyle></math>


The input can be a string, which prevents automatic simplification


1+131&equals;231add terms&equals;2·2add terms&equals;4multiply constants


The implicitmultiply option allows short-hand for string input.


2.1yx24.3xy&equals;2.1x24.3xdivide out common terms&equals;2.100000000x4.3divide out common terms&equals;0.4883720930xdivide constants




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


For more information on Maple 18 changes, see Updates in Maple 18.

See Also