I am stuck with these. No idea how to do them. Here is my first problem:
x^3+x^2y+xy^2+y^3 / x+y
I don't want an answer. I want an explanation on how to do it.
You can do it by long division, or you could simply note:
$\displaystyle x^3+x^2y+xy^2+y^3 $
$\displaystyle = (x^3 + x^2y) + (xy^2 + y^3)$
$\displaystyle = x^2(x + y) + y^2(x + y)$
$\displaystyle = (x^2 + y^2)(x + y)$
Thus
$\displaystyle \frac{x^3+x^2y+xy^2+y^3}{x + y} = x^2 + y^2$
Since this is a division of a linear polynomial in x you could also rig a synthetic division for it.
-Dan