TITLE: Help me solve this polynomial problem
QUESTION [4 upvotes]: 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!

REPLY [1 votes]: Divide with a remainder:
$$
\frac{x^3+ax^2+bx+c}{x-1}= x^2+b'x+c' +\frac{p}{x-1}  
$$
and similarly for other summands. Then your question is reduced to the next one: when
$$
\frac{p}{x-1}  +\frac{q}{x-2}  +\frac{r}{x-3}  
$$
is a polynomial? The answer: only for $p=q=r=0$, since the quadratic polynomial 
$$
p(x-2)(x-3)+q(x-1)(x-3)+r(x-2)(x-1)
$$
cannot be divided by $(x-1)(x-2)(x-3)$. Hence $x^3+ax^2+bx+c$ is divided by $x-1$ etc.
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