298. An unknown polynomial gives a remainder of 2 when divided by $x-1$, and a remainder of 1 when divided by $x-2$. What remainder does this polynomial give when divided by $(x-1)(x-2)$?
298. Let $p(x)$ be our unknown polynomial, $q(x)$ the quotient from dividing this polynomial by $(x-1)(x-2)$, and $r(x)=a x+b$ the sought remainder:

$$
p(x)=(x-1)(x-2) q(x)+a x+b
$$

According to the problem, we have:

$$
\begin{aligned}
& p(x)=(x-1) q_{1}(x)+2, \text { hence } p(1)=2 \\
& p(x)=(x-2) q_{2}(x)+1, \text { hence } p(2)=1
\end{aligned}
$$

Substituting $x=1$ and $x=2$ into the equation (*), we get:

$$
\begin{aligned}
& 2=p(1)=a+b \\
& 1=p(2)=2 a+b
\end{aligned}
$$

from which

$$
a=-1, \quad b=3
$$

Thus, the sought remainder is $-x+3$.