Question 22, Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of the function. \[ f(x)=5 x^{4}+2 x^{2}+4 x-7 \] The function $f(x)$ has positive zero(s). (Type a whole number.)
Solution
Step 1 ：The function is a polynomial of degree 4. The number of positive zeros of a polynomial function can be determined by the number of sign changes in the function's coefficients when the function is written in standard form. The function in standard form is \(f(x)=5x^4 + 2x^2 + 4x - 7\). The coefficients are 5, 2, 4, and -7. There are two sign changes (from 4 to -7), so the function has either 2 or 0 positive zeros. We can't determine the exact number without further calculations.
Step 2 ：By solving the equation, we find the roots of the function. The roots are complex numbers, which are not real numbers. However, we are only interested in the positive roots.
Step 3 ：After filtering out the non-positive roots, we find that there are 3 positive roots.
Step 4 ：Final Answer: The function \(f(x)\) has \(\boxed{3}\) positive zero(s).
