### Instruction: A polynomial with integer coefficients is of the form
\[3x^3 + a_2 x^2 + a_1 x - 6 = 0.\]Enter all the possible integer roots of this polynomial, separated by commas.### Response: To find the possible integer roots of a polynomial with integer coefficients, I can use the rational root theorem.
This theorem says that if p/q is a rational root of the polynomial, where p and q are integers with no common factors, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
In this case, the constant term is -6 and the leading coefficient is 3.
The factors of -6 are $\pm 1, \pm 2, \pm 3, \pm 6$, and the factors of 3 are $\pm 1, \pm 3$.
Therefore, the possible rational roots are $\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}$.
However, the problem asks for the possible integer roots, not the rational roots.
So I can ignore the fractions and only consider the integers.
The possible integer roots are $\pm 1, \pm 2, \pm 3, \pm 6$.