Wolfram Language & System 11.0 (2016)|Legacy Documentation

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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.

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ExpandNumerator writes the numerator of each term in expanded form.
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Expand expands the numerator of each term, and divides all the terms by the appropriate denominators.
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ExpandDenominator expands out the denominator of each term.
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ExpandAll does all possible expansions in the numerator and denominator of each term.
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ExpandAll[expr,patt], etc.avoid expanding parts which contain no terms matching patt

Controlling expansion.

This avoids expanding the term which does not contain z.
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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.

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Together puts all terms over a common denominator.
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You can use Factor to factor the numerator and denominator of the resulting expression.
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Apart writes the expression as a sum of terms, with each term having as simple a denominator as possible.
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Cancel cancels any common factors between numerators and denominators.
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Factor first puts all terms over a common denominator, then factors the result.
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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.
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This gives the partial fraction decomposition with respect to x.
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Here is the partial fraction decomposition with respect to y.
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