5. For what values of $a$, $b$, and $c$ is the polynomial $x^{4}+a x^{2}+b x+c$ divisible by $(x-1)^{3}$?
[Solution] The polynomial $x^{4}+a x^{2}+b x+c$ divided by $x-1$ yields the quotient
$$
\begin{array}{l}
P_{1}(x)=x^{3}+x^{2}+(a+1) x+a+b+1 \text { and the remainder } \\
Q_{1}(x)=a+b+c+1 .
\end{array}
$$

By the problem, $Q_{1}(x)=a+b+c+1=0$.
Also, $P_{1}(x)$ divided by $x-1$ yields the quotient
$$
P_{2}(x)=x^{2}+2 x+(a+3) \text { and the remainder } Q_{2}(x)=2 a+b+4 \text {. }
$$

By the problem, $Q_{2}(x)=2 a+b+4=0$.
Also, $P_{2}(x)$ divided by $x-1$ yields the quotient
$$
P_{3}(x)=x+3 \text { and the remainder } Q_{3}(x)=a+6 \text {. }
$$

By the problem, $Q_{3}(x)=a+6=0$,
so $a=-6$.
Substituting into (2) gives $b=8$.
Substituting again into (1) gives $c=-3$.