The coefficient of x^{19} in the polynomial ( | vedclass

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(B) The given polynomial is $P(x) = (x-2^0)(x-2^1)(x-2^2) \dots (x-2^{19})$.This is a product of $20$ factors of the form $(x-a_i)$,where $a_i = 2^i$

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$-(2^{20} - 1)$

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(B) The given polynomial is $P(x) = (x-2^0)(x-2^1)(x-2^2) \dots (x-2^{19})$.This is a product of $20$ factors of the form $(x-a_i)$,where $a_i = 2^i$ for $i = 0, 1, 2, \dots, 19$.When we expand this product,the term $x^{19}$ is obtained by choosing $x$ from $19$ factors and the constant term $-a_i$ from the remaining factor.Thus,the coefficient of $x^{19}$ is the sum of the constant terms with a negative sign:$	ext{Coeff. of } x^{19} = -(2^0 + 2^1 + 2^2 + \dots + 2^{19})$.This is a geometric series with $n=20$ terms,first term $a=1$,and common ratio $r=2$.The sum is given by $S_n = \frac{a(r^n - 1)}{r - 1}$.$	ext{Sum} = \frac{1(2^{20} - 1)}{2 - 1} = 2^{20} - 1$.Therefore,the coefficient of $x^{19}$ is $-(2^{20} - 1)$.

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

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