lean_workbook_plus_61807

Let $P(x)$ be a linear function, $P(x)=ax+b$ . Then, if this works out, our answer will be just $a^2+b^2$ , because the remainder when a linear function $P(x)$ is divided by a quadratic is just $P(x)$ . Now, using synthetic division, the remainder when $ax+b$ is divided by $x - 2$ is $b + 2a = 3$ . Similarly, for $x-3$ , the remainder is $b + 3a = 2$ . Solving, we get $a = -1$ and $b = 5$ , so the function is $P(x) = -x + 5.$ Success! Our answer is $(-1)^2+5^2=\boxed{26}$ .