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Find coefficients of $x^{2012}$ in $(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024)$

Attempt: i have break $2012$ in to sum of power of $2$

as $2012 = 2^{10}+2^{9}+2^{8}+2^7+2^6+2^4+2^3+2^2$

but wan,t be able to go further, could some help me , thanks

DXT
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    Each term is formed by taking exactly $1$ term from each bracket, and then multiplying all of the terms taken. – DHMO Apr 13 '17 at 18:35
  • @ DHMO i did not understand that, please explain me, thanks – DXT Apr 13 '17 at 18:36
  • For example, what is the coefficient of $x^5$ in $(x+1)(x^2+2)(x^4+4)$? – DHMO Apr 13 '17 at 18:41
  • @ DHMO it is $2$,but i did not understand – DXT Apr 13 '17 at 18:45
  • The only route to $x^{2012}$ is to take the powers of $x$ you discovered, which means you need to take the constant terms from the other polynomials. – Joffan Apr 13 '17 at 19:05
  • Continuinng with DHMO's example: in order to get a term containing $x^5$, you needed to choose all combinations of terms from each factor that contribute to making up the $x^5$ term, e.g. $x$ from $x+1$, $2$ from $x^2+2$, $x^4$ from $x^4+4$, the product of which gives $x\cdot2\cdot x^4=2x^5$, and hence you get a coefficient of $2$. This is the only way to get a term with $x^5$, so you're done. – user170231 Apr 13 '17 at 19:07

2 Answers2

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$2012 = 2047 - 35 = (\sum_{k=0}^{10}{2^k}) - 35$

$35 = 2^5+2^1+2^0 => 2012 = 2^{10}+2^9+2^8+2^7+2^6+2^4+2^3+2^2$

So this means that the for the parenthesis with the power of $x$ in this set: ${10, 9, 8, 7, 6, 4, 3, 2}$, the part with $x$ is multiplied. So for the rest of the parenthesis, the number is chosen. So the coefficient of $x^{2012}$ is: $2^5*2^1*2^0 = 2^6 = 64$

Soroush khoubyarian
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What is the coefficient of $x^5$ in $(x+1)(x^2+2)(x^4+4)(x^8+8)(x^{16}+16)$?

$5$ in binary is $101_2$, or $5 = 2^0 + 2^2$

This means the coefficient of $x^5$ will be formed by looking at the term $x^5 = x^{2^0}x^{2^2} = x \cdot x^4$ and multiplying it by the constant term contributed from everything else in the expression.

$(x+1)(x^2+2)(x^4+4)(x^8+8)(x^{16}+16) \\= \left[(x+1)(x^4+4)\right](x^2+2)(x^8+8)(x^{16}+16) \\= \left[x^5 + 4x + x^4 + 4\right](x^2+2)(x^8+8)(x^{16}+16)$

Note that now the answer only depends on taking $x^5$ times the constant term from the righthand part, i.e. $2^1 \cdot 2^3 \cdot 2^4 = 2 \cdot 8 \cdot 16 = 256$

In other words: Multiply together the constant terms corresponding to the pieces that aren't involved in the binary representation of the exponent you want.

We know that $x^{2012} = x^{2^2}x^{2^3}x^{2^4}x^{2^6}x^{2^7}x^{2^8}x^{2^9}x^{2^{10}}$, therefore the coefficient of $x^{2012}$ is $2^0 \cdot 2^1 \cdot 2^5 = 64$.

Marcus Andrews
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