### 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 can use the rational root theorem.
This theorem says that if a rational root of the form p/q exists, where p and q are integers with no common factors, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
In this case, the constant term is -11 and the leading coefficient is 1, so p must be a factor of -11 and q must be a factor of 1.
The factors of -11 are -1, 1, -11, and 11, and the factors of 1 are -1 and 1.
Therefore, the possible rational roots are -1/1, 1/1, -11/1, and 11/1, which simplify to -1, 1, -11, and 11.
These are all the possible integer roots of the polynomial, since any other rational root would have a denominator other than 1.