1. Find for which values of the real number $a$ all the roots of the polynomial, in the variable $x$,

$$
x^{3}-2 x^{2}-25 x+a
$$

are integers.
Solution 1. Let $\alpha, \beta$ and $\gamma$ be the roots of the polynomial. Applying the Cardano-Vieta formulas, we have

$$
\alpha+\beta+\gamma=2, \quad \alpha \beta+\alpha \gamma+\beta \gamma=-25
$$

Now,

$$
\alpha^{2}+\beta^{2}+\gamma^{2}=(\alpha+\beta+\gamma)^{2}-2(\alpha \beta+\alpha \gamma+\beta \gamma)=54
$$

Since $\alpha, \beta$ and $\gamma$ are integers, we seek integer solutions to the system of equations

$$
\alpha+\beta+\gamma=2, \quad \alpha^{2}+\beta^{2}+\gamma^{2}=54
$$

From the second equation, we see that the only possible solutions are

$$
( \pm 1, \pm 2, \pm 7), \quad( \pm 2, \pm 5, \pm 5), \quad( \pm 3, \pm 3, \pm 6)
$$

Considering the first equation, the only possible solution is $(2,5,-5)$, and thus $a=50$.