Final answer:
The polynomial function f(x) = x^4 - 2x^3 - 6x^2 + 6x + 9 has 4 total zeros, a list of possible rational zeros that can be determined using the p/q method, and a guideline provided by Descartes' Rule of Signs to estimate the number of positive, negative, and complex zeros. The actual zeros would be found through techniques like synthetic division or the quadratic formula for any reducible parts of the polynomial.
Step-by-step explanation:
For the function f(x) = x^4 - 2x^3 - 6x^2 + 6x + 9:
A) The function, being a polynomial of degree 4, has 4 total zeros. This includes real and complex zeros.
B) To list all possible rational zeros, we use the p/q method where p are factors of the constant term and q are factors of the leading coefficient. The possible rational zeros for this function using p/q are ±1, ±3, ±9.
C) According to Descartes' Rule of Signs, the function has 2 or 0 positive zeros and 2 or 0 negative zeros. There are no sign changes for f(-x), so no negative real zeros are possible. Therefore, all remaining zeros must be complex.
D) To find all the zeros of the function, one would typically use techniques such as synthetic division, factoring, or the Rational Root Theorem to test the possible rational zeros, which could potentially reduce the polynomial to a quadratic form. For any resulting quadratic equation, we could then use the quadratic formula, ax² + bx + c = 0, to solve for the zeros. Though the specific zeros are not shown here, this is the process one would follow.
It is essential to eliminate terms wherever possible to simplify the algebra, and to check the answer for reasonableness afterward.