Lösung 2.1:5a

Aus Online Mathematik Brückenkurs 1

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In the same way that we calculated fractions, we can subtract the terms' numerators if we first
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In the same way that we calculated fractions, we can subtract the terms' numerators if we first expand the fractions so that they have the same denominator. Because the denominators are <math>x-x^{2}=x(1-x)</math> and <math>x</math>, the lowest common denominator is <math>x(1-x)</math>,
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expand the fractions so that they have the same denominator. Because the denominators are
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<math>x-x^{2}=x\left( 1-x \right)</math>
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and
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<math>x</math>, the lowest common denominator is
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<math>x\left( 1-x \right)</math>:
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{{Displayed math||<math>\begin{align}
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\frac{1}{x-x^{2}}-\frac{1}{x\vphantom{x^2}}
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&= \frac{1}{x-x^{2}}-\frac{1}{x\vphantom{x^2}}\cdot \frac{1-x}{1-x\vphantom{x^2}}\\[5pt]
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&= \frac{1}{x-x^{2}}-\frac{1-x}{x-x^{2}}\\[5pt]
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&= \frac{1-(1-x)}{x-x^{2}}\\[5pt]
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&= \frac{1-1+x}{x-x^{2}}\\[5pt]
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&= \frac{x}{x-x^{2}}\,\textrm{.}
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\end{align}</math>}}
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<math>\begin{align}
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This fraction can be simplified by eliminating the factor ''x'' from the numerator and denominator
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& \frac{1}{x-x^{2}}-\frac{1}{x}=\frac{1}{x-x^{2}}-\frac{1}{x}\centerdot \frac{1-x}{1-x}=\frac{1}{x-x^{2}}-\frac{1-x}{x-x^{2}} \\
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& \\
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& =\frac{1-\left( 1-x \right)}{x-x^{2}}=\frac{1-1+x}{x-x^{2}}=\frac{x}{x-x^{2}} \\
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& \\
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\end{align}</math>
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{{Displayed math||<math>\frac{x}{x-x^{2}} = \frac{x}{x(1-x)} = \frac{1}{1-x}\,\textrm{.}</math>}}
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This fraction can be simplified by eliminating the factor
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<math>x</math>
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from the numerator and denominator.
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<math>\begin{align}
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& \frac{x}{x-x^{2}}=\frac{x}{x\left( 1-x \right)}=\frac{1}{1-x}. \\
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& \\
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\end{align}</math>
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Version vom 10:35, 23. Sep. 2008

In the same way that we calculated fractions, we can subtract the terms' numerators if we first expand the fractions so that they have the same denominator. Because the denominators are \displaystyle x-x^{2}=x(1-x) and \displaystyle x, the lowest common denominator is \displaystyle x(1-x),

Vorlage:Displayed math

This fraction can be simplified by eliminating the factor x from the numerator and denominator

Vorlage:Displayed math

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_2.1:5a“