Final answer:
The polynomial has 16 possible rational roots.
Step-by-step explanation:
A polynomial can have possible rational roots which can be found using the rational root theorem. The rational root theorem states that if a polynomial has a rational root, it 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 -30 and the leading coefficient is 6. The factors of 6 are 1, 2, 3, and 6. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. Therefore, the possible rational roots are +/- 1, 2, 3, 5, 6, 10, 15, and 30.
So, the polynomial 6x⁴ - 11x³ + 8x² - 33x - 30 has a total of 16 possible rational roots.