Aus Online Mathematik Brückenkurs 1

The denominators \displaystyle x-1 and \displaystyle x^{2} do not have a common denominator, so the lowest common denominator is \displaystyle x^{2}\left( x-1 \right). We treat all three terms so that they have a common denominator and then start simplifying:

\displaystyle \begin{align} & x+\frac{1}{x-1}+\frac{1}{x^{2}}=x\centerdot \frac{x^{2}\left( x-1 \right)}{x^{2}\left( x-1 \right)}+\frac{1}{x-1}\centerdot \frac{x^{2}}{x^{2}}+\frac{1}{x^{2}}\centerdot \frac{x-1}{x-1} \\ & =\frac{x^{3}\left( x-1 \right)+x^{2}+\left( x-1 \right)}{x^{2}\left( x-1 \right)}=\frac{x^{4}-x^{3}+x^{2}+x-1}{x^{2}\left( x-1 \right)} \\ \end{align}