Question

The coefficient of $x^{19}$ in the polynomial $(x-1)(x-2)\left(x-2^2\right)\dots \left(x-2^{19}\right)$ is

• A. $2^{20}-2^{19}$
• B. $1-2^{20}$
• C. $2^{20}$
• D. None of these

Answer 1

Answer: (B)

The coefficient of $x^{19}$ in the polynomial
$(x-1)(x-2)\left(x-2^2\right)\dots \left(x-2^{19}\right)$ is
$-\left(1+2+2^2+\dots .+2^{19}\right)$
$=-1\left(\frac{2^{20}-1}{2-1}\right)=1-2^{20}$