### 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, which says that if a rational number p/q is a root, 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 -11, -1, 1, and 11, and the factors of 1 are -1 and 1, so the possible rational roots are
\[\frac{-11}{-1}, \frac{-11}{1}, \frac{-1}{-1}, \frac{-1}{1}, \frac{1}{-1}, \frac{1}{1}, \frac{11}{-1}, \frac{11}{1}.\]
Simplifying these fractions, I get the possible rational roots as
\[-11, 11, 1, -1, -1, 1, -11, 11.\]
However, some of these are repeated, so I can eliminate the duplicates and write the possible rational roots as
\[-11, 11, 1, -1.\]