<s>[INST] Polynomial problem with unknown coefficients $a, b, c$ $p(x)=x^3 +ax^2+bx+c$, where $a,b,c$ are distinct non-zero integers. Suppose $p(a)=a^3$ and $p(b)=b^3$, find $p(13)$.
Ok so I've gotten $a^3-b^3=a^3-b^3+a^3-ab^2+ab-b^2$, 
So $(a-b)[a(a+b)+b]=0$, so $b=-\frac{a^2}{a+1}$ and then getting $c=-\frac{a^4}{a+1}$ does not help much I guess. Will someone tell me how I should be thinking about this question? Thanks.
 [/INST] Put $n=a+1$, then: $$b={-a^2\over a+1} = -{(n-1)^2\over n} = -{n^2-2n+1\over n} = -n+2-{1\over n} \in\mathbb{Z}$$
So $$n\mid 1\implies n =\pm 1 \implies a=0 \vee a= -2 $$
Now if $a=0$ then $b=0$ and this can't be, so $a=-2$ and $b=4$ and $c= 16$.
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