Why is the difference between these two functions a constant?

f(x)=2x2−xx2−x+1

g(x)=x−2x2−x+1Since the

denominatorsare equal and thenumeratorsdiffer 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

intuitiveexplanation of what is going on here? Perhaps using agraphof 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(x2−x+1)2

As a solution I found:

23√3arctan(2x−1√3)+2x2−x3(x2−x+1)+C

Whereas my calculusbook gave as the solution:

23√3arctan(2x−1√3)+x−23(x2−x+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*