Verified answer

By the remainder theorem, the remainder will be

P(2) = 2^2011 - 2*2^2010 + 1

= 2^2011 - 2^2011 + 1

= 1.

Or, if the remainder theorem doesn't leap to mind, you can see that x^2011 - 2x^2010 factors into x^2010(x-2), so the remainder is 1.