Question

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

• A. $1-{20}^{20}$
• B. $2+{20}^{20}$
• C. $4-{20}^{20}$
• D. $Noneofthese$

Answer 1

Answer: (A)

$1-{20}^{20}$

$(x-1)(x-2)(x-2^2)....(x-2^{19})$

Since above is factdred on $x$

Now, highest power of $x=30$

$\therefore$ coefficient of $x^{20}=\mathrm{1....}$observation

$\therefore$ Sum of speed roots of abpve equation

$=1+2+2^2+2^3+2^4....+2^{19}$

$=2^0+2^1+1^2+2^3{+}^4........+2^{19}$

$=\frac{1(2^{20}-1)}{2-1}....\left(\frac{1\left(r^n\right)-1}{r-1}\right)$

$={20}^{20}-1$

$\therefore -\frac{\left(coefficientof{x}^{19}\right)}{\left(coefficientof{x}^{20}\right)}=2^{20}-1$

$\therefore$ now coefficient of $x^{20}=1$

$\therefore$ coeffiecient of $x^{19}=1-2^{20}$