State the number of positive real zeros, negative real zeros, and imaginary zeros for g(x)= 9x^3 - 7x^2 + 10x - 4.
The only thing I know is that there are 3 sign changes.
To determine the number of positive real zeros, negative real zeros, and imaginary zeros for the given polynomial function g(x) = 9x^3 - 7x^2 + 10x - 4, you can use the Descartes' Rule of Signs and the Fundamental Theorem of Algebra.
1. Descartes' Rule of Signs:
By applying Descartes' Rule of Signs, you analyze the number of sign changes in the coefficients of the polynomial equation.
To count the number of positive real zeros:
- Keep the signs of the coefficients the same.
- Count the number of sign changes in the coefficients or write the coefficients in a row and count the number of times the signs change.
In the given polynomial g(x) = 9x^3 - 7x^2 + 10x - 4, there are 3 sign changes.
From the number of sign changes, you can determine that there are either 3 or 1 positive real zeros.
2. Fundamental Theorem of Algebra:
According to the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n complex zeros (real or imaginary) counting multiplicities.
In the case of g(x) = 9x^3 - 7x^2 + 10x - 4, the degree of the polynomial is 3. Therefore, we expect to find three complex zeros, either real or imaginary.
From the information obtained using Descartes' Rule of Signs and the Fundamental Theorem of Algebra, we can conclude:
- The number of positive real zeros is either 3 or 1.
- The number of negative real zeros is 0 or an even number.
- The number of imaginary zeros is the difference between the total number of zeros (3) and the number of real zeros.
Unfortunately, with the information given, we cannot determine the exact number of positive real zeros, negative real zeros, or imaginary zeros.