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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.

Here is a rational expression.
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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 .
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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.

Here is a rational expression.
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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 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 .
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Here is the partial fraction decomposition with respect to .
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