19. Find the remainder when $(x-1)^{100}+(x-2)^{200}$ is divided by $x^{2}-3 x+2$.
19. Answer: 1
By the division algorithm, we have
$$
(x-1)^{100}+(x-2)^{200}=\left(x^{2}-3 x+2\right) q(x)+a x+b,
$$
where $q(x)$ is the quotient, and $a x+b$ is the remainder with constants $a$ and $b$.
Note that $x^{2}-3 x+2=(x-1)(x-2)$. Therefore
$$
(x-1)^{100}+(x-2)^{200}=(x-1)(x-2) q(x)+a x+b,
$$

Putting $x=1$, we obtain
$$
(1-2)^{200}=a+b \text {, or } a+b=1 \text {. }
$$

Putting $x=2$, we obtain
$$
(2-1)^{100}=2 a+b, \text { or } 2 a+b=1 \text {. }
$$

Solving the two equations gives $a=0$ and $b=1$.