4. $\mathrm{P}$ is a polynomial. When $\mathrm{P}$ is divided by $x-1$, the remainder is -4 . When $\mathrm{P}$ is divided by $x-2$, the remainder is -1 . When $P$ is divided by $x-3$, the remainder is 4 . Determine the remainder when $\mathrm{P}$ is divided by $x^{3}-6 x^{2}+11 x-6$.
Solution: The remainder polynomial is simply the order two polynomial that goes through the points $(1,-4),(2,-1)$, and $(3,4): x^{2}-5$.