2. It is known that the roots of the equation
$$
x^{5}+3 x^{4}-4044118 x^{3}-12132362 x^{2}-12132363 x-2011^{2}=0
$$
are all integers. How many distinct roots does the equation have?
(A) 1 ;
(B) 2 ;
(C) 3 ;
(D) 4 ;
(E) 5 .
2. Answer: (C).
Let the roots be $n_{1} \leq n_{2} \leq n_{3} \leq n_{4} \leq n_{5}$. Then the polynomial can be factorized as $\left(x-n_{1}\right)\left(x-n_{2}\right)\left(x-n_{3}\right)\left(x-n_{4}\right)\left(x-n_{5}\right)=x^{5}-\left(n_{1}+n_{2}+n_{3}+n_{4}+n_{5}\right) x^{4}+\cdots-n_{1} n_{2} n_{3} n_{4} n_{5}$. Compare the coefficients: $n_{1}+n_{2}+n_{3}+n_{4}+n_{5}=-3$ and $n_{1} n_{2} n_{3} n_{4} n_{5}=2011^{2}$. Then $n_{1}=-2011, \quad n_{2}=n_{3}=n_{4}=-1, \quad n_{5}=2011$.