According to Descartes' Rule of Signs, the given polynomial function f(x) = 9x^3 - 4x^2 + x + 4.5 has either 2 or 0 possible positive zeros and either 1 or 2 possible negative zeros.
To use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of a polynomial function, you need to examine the signs of the coefficients in the polynomial equation.
Let's consider the given polynomial function: f(x) = 9x^3 - 4x^2 + x + 4.5
1. Count the number of sign changes in the coefficients of the polynomial function. In this case, we have two sign changes: from positive (+9) to negative (-4), and from negative (-4) to positive (+1).
2. The number of positive zeros can be determined by considering the number of sign changes. Since we have two sign changes, the number of possible positive zeros is either 2 or 0.
3. Now, consider the polynomial f(-x) to determine the number of negative zeros. In this case, f(-x) = 9(-x)^3 - 4(-x)^2 + (-x) + 4.5, which simplifies to -9x^3 - 4x^2 - x + 4.5.
Again, count the number of sign changes in the coefficients of f(-x). Here, we have one sign change: from negative (-4) to positive (+1).
4. The number of negative zeros can be determined by considering the number of sign changes in f(-x). Since we have one sign change, the number of possible negative zeros is either 1 or 0.