Intuition for why the difference between 2×2−xx2−x+1\frac{2x^2-x}{x^2-x+1} and x−2×2−x+1\frac{x-2}{x^2-x+1} is a constant?

Why is the difference between these two functions a constant?

f(x)=2x2xx2x+1
g(x)=x2x2x+1

Since the denominators are equal and the numerators differ in degree I would never have thought the difference of these functions would be a constant.

Of course I can calculate it is true: the difference is 2, but my intuition is still completely off here. So, who can provide some intuitive explanation of what is going on here? Perhaps using a graph of some kind that shows what’s special in this particular case?

Thanks!


BACKGROUND: The background of this question is that I tried to find this integral:

xdx(x2x+1)2

As a solution I found:

233arctan(2x13)+2x2x3(x2x+1)+C

Whereas my calculusbook gave as the solution:

233arctan(2x13)+x23(x2x+1)+C

I thought I made a mistake but as it turned out, their difference was constant, so both are valid solutions.

Answer

Would you be surprised that the difference of 2x2+x+1x2 and x+1x2 is 2?

Attribution
Source : Link , Question Author : GambitSquared , Answer Author : Jared Goguen

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