Processing Math: Done

Aus Online Mathematik Brückenkurs 2

We start by adding and taking away x2 in the numerator, so that, in combination with x3, we obtain the expression x3+x2=x2x+1  which can be simplified with the denominator x+1,

x+1x3+x+2=x+1x3+x2−x2+x+2=x+1x3+x2+x+1−x2+x+2=x+1x2x+1+x+1−x2+x+2=x2+x+1−x2+x+2

The term −x2 in the remaining quotient needs to complemented with −x so that we get −x2−x=−xx+1 , which is divisible by x+1,

x2+x+1−x2+x+2=x2+x+1−x2−x+x+x+2=x2+x+1−x2−x+x+12x+2=x2+x+1−xx+1+x+12x+2=x2−x+x+12x+2

The last quotient divides perfectly and we obtain

x2−x+x+12x+2=x2−x+2

A quick check of whether

x+1x3+x+2=x2−x+2

is the correct answer is to investigate whether

x3+x+2=x2−x+2x+1 

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

x2−x+2x+1=x3+x2−x2−x+2x+2=x3+x+2