### 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, 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 the factors of -6: \[\pm 1, \pm 2, \pm 3, \pm 6.\]
The possible values of q are the factors of 3: \[\pm 1, \pm 3.\]
This means that the possible rational roots are the fractions formed by dividing any p by any q, in lowest terms.
I can list them as follows: \[\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{3}, \pm \frac{2}{3}.\]
However, the problem only asks for the integer roots, so I can exclude the fractions and only keep the integers.