3.3.3 Structural Operations on Rational ExpressionsFor 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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t = (1 + x)^2 / (1 - x) + 3 x^2 / (1 + x)^2 + (2 - x)^2
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| ExpandNumerator writes the numerator of each term in expanded form. | |
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ExpandNumerator[t]
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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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ExpandDenominator[t]
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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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ExpandAll[(x + 1)^2/y^2 + (z + 1)^2/z^2, 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. | Here is a rational expression. | |
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u = (-4x + x^2)/(-x + x^2) + (-4 + 3x + x^2)/(-1 + x^2)
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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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v = (x^2+y^2)/(x + x y)
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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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