Final answer:
To determine the number of positive, negative, and nonreal complex zeros for the function f(x)=7x^4+4x^2+3x-9, we use Descartes' Rule of Signs and the Fundamental Theorem of Algebra. There are zero positive real roots, one negative real root, and three nonreal complex roots.
Step-by-step explanation:
The question asks to determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of the function f(x)=7x4+4x2+3x-9. To answer this, we can use the Descartes' Rule of Signs to find the number of positive and negative roots, and the Fundamental Theorem of Algebra for the complex roots.
Firstly, we can evaluate the number of positive real zeros by observing changes in the sign of the coefficients when x is substituted by a positive number. Next, we can find the number of negative real zeros by substituting x with -x and then observing the sign changes. Finally, since a fourth-degree polynomial has exactly four roots, the number of nonreal complex roots can be determined by subtracting the total number of real roots from four.
By applying these steps to the original polynomial: