B2. When the product
$$
\left(2021 x^{2021}+2020 x^{2020}+\cdots+3 x^{3}+2 x^{2}+x\right)\left(x^{2021}-x^{2020}+\cdots+x^{3}-x^{2}+x-1\right)
$$
is expanded and simplified, what is the coefficient of $x^{2021} ?$
Solution: Expanding terms that yield $x^{2021}$ we get
$$
\begin{aligned}
\left(2021 x^{2021}\right)(-1) & +\left(2020 x^{2020}\right)(x)+\left(2019 x^{2019}\right)\left(-x^{2}\right)+\cdots= \\
& =-2021 x^{2021}+2020 x^{2021}-2019 x^{2021}+\cdots \\
& =(-2021+[2020-2019]+[2018-2017]+\cdots+[4-3]+[2-1]) x^{2021} \\
& =(-2021+1010 \times 1) x^{2021} \\
& =-1011 x^{2021}
\end{aligned}
$$
$\therefore$ The coefficient of $x^{2021}$ is -1011 .
Answer: -1011 .