Aus Online Mathematik Brückenkurs 2

We start by adding and taking away \displaystyle x^{2} in the numerator, so that, in combination with \displaystyle x^{3}, we obtain the expression \displaystyle x^{3}+x^{2}=x^{2}\left( x+1 \right) which can be simplified with the denominator \displaystyle x+1,

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

The term \displaystyle -x^{2} in the remaining quotient needs to complemented with \displaystyle -x so that we get \displaystyle -x^{2}-x=-x\left( x+1 \right), which is divisible by \displaystyle x+1,

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

The last quotient divides perfectly and we obtain

\displaystyle x^{2}-x+\frac{2x+2}{x+1}=x^{2}-x+2.

A quick check of whether

\displaystyle \frac{x^{3}+x+2}{x+1}=x^{2}-x+2.

is the correct answer is to investigate whether

\displaystyle x^{3}+x+2=\left( x^{2}-x+2 \right)\left( x+1 \right)

holds. If we expand the right-hand side, we see that the relation really does hold:

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