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Product of Polynomials with Powers of Two as Exponents

Source: 2012 AMC 12A Problem #20

February 8, 2012
algebrapolynomialAMC

Problem Statement

Consider the polynomial P(x)=∏k=010(x2k+2k)=(x+1)(x2+2)(x4+4)⋯(x1024+1024).P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024). The coefficient of x2012x^{2012} is equal to 2a2^a. What is aa?
<spanclass=′latex−bold′>(A)</span> 5<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 7<spanclass=′latex−bold′>(D)</span> 10<spanclass=′latex−bold′>(E)</span> 24 <span class='latex-bold'>(A)</span>\ 5\qquad<span class='latex-bold'>(B)</span>\ 6\qquad<span class='latex-bold'>(C)</span>\ 7\qquad<span class='latex-bold'>(D)</span>\ 10\qquad<span class='latex-bold'>(E)</span>\ 24
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