### Instruction: A polynomial with integer coefficients is of the form
\[x^3 + a_2 x^2 + a_1 x - 11 = 0.\]Enter all the possible integer roots of this polynomial, separated by commas.### Response: To find the possible integer roots of this polynomial, I will 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 (-11) and q is a factor of the leading coefficient (1).
Since the leading coefficient is 1, q can only be 1 or -1.
So the possible rational roots are p/1 or p/-1, where p is a factor of -11.
The factors of -11 are 1, -1, 11, and -11.
So the possible rational roots are 1, -1, 11, -11, -1/1, 1/-1, -11/1, and 11/-1.
Simplifying, we get the same four possibilities: 1, -1, 11, and -11.
These are the only possible rational roots, and since the polynomial has integer coefficients, any integer root must also be rational.
So the possible integer roots are 1, -1, 11, and -11.