Final answer:
The possible rational roots of the polynomial function P(x) = 3x^4 - 4x^3 - x^2 - 7 are ±1, ±7, ±1/3, and ±7/3.
Step-by-step explanation:
The possible rational roots of the polynomial function P(x) = 3x^4 - 4x^3 - x^2 - 7 can be found using the Rational Root Theorem.
This theorem states that if a rational number P/Q (where P is an integer factor of the constant term and Q is an integer factor of the leading coefficient) is a root of the polynomial, then P/Q is a possible rational root.
In this case, the constant term is -7 and the leading coefficient is 3. The factors of -7 are 1, -1, 7, and -7, and the factors of 3 are 1 and 3. So, the possible rational roots are: