Polynomial remainders I was trying some problems from this problem seminar worksheet and I was unable to make much progress on this particular problem. 
Find the remainder when $(x+2)^{2011} - (x+1)^{2011}$ is divided by $x^2+x+1$

Hint:
One approach would be an analogue to what you do when reducing in modular arithmetic.
If, say, you were computing $5^{2011}-4^{2011}\pmod{31}$, you might first rewrite $r^{2011}=5\cdot 5^{2010}=5\cdot 5^{3\cdot 670}=5\cdot(125)^{670}$ and reduce $125$ using $31$, then repeat this process.
So in this case, you might try $(x+2)^{2011}\\=(x+2)(x+2)^{2\cdot 1005}\\=(x+2)(x^2+4x+4)^{1005}\\\equiv(x+2)(3x+3)^{1005}\pmod{x^2+x+1})$
Then you could repeat the same process on $(3x+3)^{1005}=(3x+3)(9x^2+18x+9)^{502}\equiv(3x+3)(9x)^{502}$
There are also probably smarter ways to reduce the expression more quickly by noticing a clever factorization.
