For example, what values of $9 a, b, c$ make the polynomial $x^{4}+a x^{2}+b x+c$ divisible by $(x-1)^{3}$?
(Kyiv Olympiad problem)
Solve the polynomial $x^{4}+a x^{2}+b x+c$ divided by $x-1$ to get the quotient
$P_{1}(x)=x^{3}+x^{2}+(a+1) x+a+b+1$ and the remainder $Q_{1}(x)=a+b+c+1$.
From the problem, we know $Q_{1}(x)=a+b+c+1=0$.
Also, $P_{1}(x)$ divided by $x-1$ gives the quotient
$P_{2}(x)=x^{2}+2 x+(a+3)$ and the remainder $Q_{2}(x)=2 a+b+4$.
From the problem, we know $Q_{2}(x)=2 a+b+4=0$.
Again, $P_{2}(x)$ divided by $x-1$ gives the quotient
$P_{3}(x)=x+3$ and the remainder $Q_{3}(x)=a+b$.
From the problem, we know $Q_{3}(x)=a+6=0$, which means $a=-6$.
Substituting $a=-6$ into (2), we get $b=8$.
Substituting $b=8, a=-6$ into (1), we get $c=-3$.
Therefore, $a=-6, b=8, c=-3$ are the solutions.