<s>[INST] Help me solve this polynomial problem I have to find $a$,$b$ and $c$ so that this expression is a polynomial:
$$\frac{x^3+ax^2+bx+c}{x-1} + \frac{x^3+bx^2+cx+a}{x-2} + \frac{x^3+cx^2+ax+b}{x-3}$$
Please give me hints. Thanks!
 [/INST] Hint: $x^3 + ax^2 + bx + c$ must have a root at $1$. This is because $(x - 1)$ must be a divisor of $x^3 + ax^2 + bx + c$, or else there would be a term involving the quantity $\frac{1}{x - 1}$. In a similar manner, $x^3 + bx^2 + cx + a$ has a root at $2$ and $x^3 + cx^2 + ax + b$ has a root at $3$. This gives us three equations, the first of which is
$$1 + a + b + c = 0$$
Now use this to determine $a, b, c$.
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