Final answer:
The Rational Zeros Theorem allows us to list all potential rational zeros of a polynomial by dividing the factors of the constant term by the factors of the leading coefficient. For f(x) = -9x^4 - 6x^3 + 3x^2 - 4x + 1, the possible rational zeros are ±1, -1, ±3, -1/3, ±9, and -1/9.
Step-by-step explanation:
The Rational Zeros Theorem is a useful tool when we want to list all possible rational zeros for a polynomial function. For the function f(x) = -4x - 6x^3 - 9x^4 + 1 + 3x^2, we first need to rewrite it in standard form as f(x) = -9x^4 - 6x^3 + 3x^2 - 4x + 1. According to the theorem, the potential rational zeros are all the factors of the constant term (here, +1) divided by the factors of the leading coefficient (here, -9).
To find these factors, we look at the integer divisors of +1 and -9.
- Factors of +1: ±1
- Factors of -9: ±1, ±3, ±9
Combining these factors, the possible rational zeros include ±1, ±1/3, ±1/9. Remember, each of these could be positive or negative yielding a total list of possible rational zeros as: ±1, -1, ±3, -1/3, ±9, and -1/9.
After listing these possible zeros, you would typically proceed to use synthetic division or other methods to test which, if any, of these candidates are actual zeros of the polynomial.