Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of the function
\[
f(x)=5 x^{4}+4 x^{2}+4 x-8
\]
The function $f(x)$ has 1 positive zero(s)
(Type a whole number.)
The function $f(x)$ has $\square$ negative zero(s)
(Type a whole number.)
Answer
Final Answer: The function $f(x)$ has \(\boxed{1}\) negative zero and \(\boxed{2}\) nonreal complex zeros.
Step 1 ：The function $f(x)$ is a polynomial of degree 4, so it has 4 zeros in the complex plane.
Step 2 ：The zeros of a polynomial with real coefficients always come in conjugate pairs. Therefore, the number of nonreal complex zeros must be even.
Step 3 ：We know that there is 1 positive zero.
Step 4 ：There must be 1 negative zero to balance out the positive zero.
Step 5 ：The remaining 2 zeros must be nonreal complex zeros.
Step 6 ：Final Answer: The function $f(x)$ has \(\boxed{1}\) negative zero and \(\boxed{2}\) nonreal complex zeros.
