### 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 of the form p/q must satisfy p dividing the constant term and q dividing the leading coefficient.
Since the constant term is -6 and the leading coefficient is 3, the possible values of p are the factors of -6, which are +-1, +-2, +-3, and +-6, and the possible values of q are the factors of 3, which are +-1 and +-3.
Therefore, the possible rational roots are +-1, +-2, +-3, +-6, and +-1/3, +-2/3, +-3/3, and +-6/3.
Simplifying, I get +-1, +-2, +-3, +-6, and +-1/3, +-2/3, and +-2 as the possible rational roots.
However, the problem only asks for the possible integer roots, so I can eliminate the fractions and only keep the integers.
The possible integer roots are +-1, +-2, +-3, and +-6.