Find the coefficient of $x^{19}$ in the polynomial $(x - 1) (x - 2) (x - 2^{2}) . . . . . . . . . . . . (x - 2^{19})$

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Solution

Given the polynomial,
$(x - 1) (x - 2) (x - 2^{2}) . . . . . . . . . . . . (x - 2^{19})$
$= (x - 2^{0}) (x - 2^{1}) (x - 2^{2}) . . . . . . . . . . . . (x - 2^{19})$
$= x^{20} - x^{19} (1 + 2 + 2^{2} + . . . . . + 2^{19}) + . . . .$
So it is clear that the co-efficient of $x^{19}$ is
$- (1 + 2 + 2^{2} + . . . . . + 2^{19})$
$= - \frac{2^{20} - 1}{2 - 1}$
$= 1 - 2^{20}$ .

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