### 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, I can use the rational root theorem, which says that any rational root p/q must have p as a factor of the constant term and q as a factor of the leading coefficient.
In this case, the constant term is -6 and the leading coefficient is 3, so the possible values of p are \[\pm 1, \pm 2, \pm 3, \pm 6\] and the possible values of q 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 question asks for integer roots only, so I can eliminate the fractions and just keep the integers.
Thus, the possible integer roots are \[\pm 1, \pm 2, \pm 3, \pm 6.\]