### 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 this polynomial, I can use the rational root theorem, which says that any rational root must be of the form p/q, where p is a factor of the constant term and q is 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 ±1, ±2, ±3, and ±6, and the possible values of q are ±1 and ±3.
This means that the possible rational roots are ±1, ±2, ±3, ±6, ±1/3, and ±2/3.
However, the question asks for the possible integer roots, which means I can ignore the fractions and only consider the integers.
So, the possible integer roots of this polynomial are ±1, ±2, ±3, and ±6.