2012 AMC 12A Problems/Problem 20

Every term in the polynomial is formed from the multiplication of one thing from every term, either x to some power or an integer.

Every number, including 2012, has a unique representation by the sum of powers of two, and that representation can be showed by converting a number to its binary form. 2012 base 10 = 11111011100 base 2, meaning 2012 = 1024 + 512 + 256 + 128 + 64 + 16 + 8 + 4.

Thus, the x^2012 term was made by multiplying x^1024 from the (x^1024 + 1024) factor, x^512 from the (x^512 + 512) factor, and so on. The only numbers not used are 32, 2, and 1.

Thus, from the (x^32 + 32), (x^2+2), (x+1) factors, 32, 2, and 1 were chosen as opposed to x^32, x^2, and x.

Thus, the coefficient of the x^2012 term is 32 * 2 * 1 = 64 = 2^6. So, 6 is the right answer, and choice B was 6.