<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] Since $b=-\frac{a^2}{a+1}$, we see that
$$c=-a^3-ab=-a^3+\frac{a^3}{a+1}=-\frac{a^4}{a+1}$$ and since $gcd(a^4,a+1)=1,$ we obtain:
$a+1\in\{-1,1\}$ and since $a\neq0$, we obtain $a=-2$, $c=16$, $b=4$,
$$f(x)=x^3-2x^2+4x+16$$ and
$$f(13)=13^3-2\cdot13^2+4\cdot13+16=1927.$$
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