Structural Operations on Rational Expressions

For ordinary polynomials, Factor and Expand give the most important forms. For rational expressions, there are many different forms that can be useful.

ExpandNumerator[expr]expand numerators only
ExpandDenominator[expr]expand denominators only
Expand[expr]expand numerators, dividing the denominator into each term
ExpandAll[expr]expand numerators and denominators completely

Different kinds of expansion for rational expressions.

Here is a rational expression.
In[1]:=
Click for copyable input
Out[1]=
ExpandNumerator writes the numerator of each term in expanded form.
In[2]:=
Click for copyable input
Out[2]=
Expand expands the numerator of each term, and divides all the terms by the appropriate denominators.
In[3]:=
Click for copyable input
Out[3]=
ExpandDenominator expands out the denominator of each term.
In[4]:=
Click for copyable input
Out[4]=
ExpandAll does all possible expansions in the numerator and denominator of each term.
In[5]:=
Click for copyable input
Out[5]=
ExpandAll[expr,patt], etc.avoid expanding parts which contain no terms matching patt

Controlling expansion.

This avoids expanding the term which does not contain .
In[6]:=
Click for copyable input
Out[6]=
Together[expr]combine all terms over a common denominator
Apart[expr]write an expression as a sum of terms with simple denominators
Cancel[expr]cancel common factors between numerators and denominators
Factor[expr]perform a complete factoring

Structural operations on rational expressions.

Here is a rational expression.
In[7]:=
Click for copyable input
Out[7]=
Together puts all terms over a common denominator.
In[8]:=
Click for copyable input
Out[8]=
You can use Factor to factor the numerator and denominator of the resulting expression.
In[9]:=
Click for copyable input
Out[9]=
Apart writes the expression as a sum of terms, with each term having as simple a denominator as possible.
In[10]:=
Click for copyable input
Out[10]=
Cancel cancels any common factors between numerators and denominators.
In[11]:=
Click for copyable input
Out[11]=
Factor first puts all terms over a common denominator, then factors the result.
In[12]:=
Click for copyable input
Out[12]=

In mathematical terms, Apart decomposes a rational expression into "partial fractions".

In expressions with several variables, you can use Apart[expr,var] to do partial fraction decompositions with respect to different variables.

Here is a rational expression in two variables.
In[13]:=
Click for copyable input
Out[13]=
This gives the partial fraction decomposition with respect to .
In[14]:=
Click for copyable input
Out[14]=
Here is the partial fraction decomposition with respect to .
In[15]:=
Click for copyable input
Out[15]=