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Aus Online Mathematik Brückenkurs 2

Imagine for a moment taking away all the terms in the numerator apart from x3. If we are to make x3 divisible by the denominator x2+3x+1, we need to add and subtract 3x2+x in order to obtain the expression x3+3x2+x=xx2+3x+1 ,

x2+3x+1x3+2x2+1=x2+3x+1x3+3x2+x−3x2−x+2x2+1=x2+3x+1x3+3x2+x+x2+3x+1−3x2−x+2x2+1=x2+3x+1xx2+3x+1+x2+3x+1−x2−x+1=x+x2+3x+1−x2−x+1

Now, we carry out the same procedure with the new quotient. To the term −x2, we add and subtract −3x−1 and obtain

x+x2+3x+1−x2−x+1=x+x2+3x+1−x2−3x−1+3x+1−x+1=x+x2+3x+1−x2−3x−1+x2+3x+13x+1−x+1=x−1+2x+2x2+3x+1