Question

Coefficient of x${}^{19}$ in the polynomial $(x-1)(x-2)..........(x-20)$ is equal to

Answer 1

The Given equation is

$(x-1)(x-2)(x-3)....(x-19)(x-20)$

Consider the first 2 factors $(x-1)$ and $(x-2)$

$(x-1)(x-2)=x^2-(1+2)x^1+(1\times 2)$

Similarly, for the rest of the terms, we can write ,

$(x-1)(x-2)(x-3)....(x-19)(x-20)$

$=x^{20}-(1+2+3+.......+19+20)x^{19}+(1\times 2+2\times 3+......19\times 20)x^{18}$

$\therefore$ Coefficient of $x^{19}=-(1+2+3+..........+19+20)=-\left[\frac{20}{2}\right(1+20\left)\right]=-10\times 21=-210$