Question 2 (1 point) State the number of complex zeros and the possible number of real and imaginary zeros for each function. $$ f(x)=x^{4}-9 x^{2}+18 $$ a \# of complex zeros: 4 Possible \# of real zeros: 4,2 , or 0 Possible \# of imaginary zeros: 4,2 , or 0 b \# of complex zeros: 6 Possible \# of real zeros: 4,2 , or 0 Possible \# of imaginary zeros: 4,2 , or 0 c \# of complex zeros: 8 Possible \# of real zeros: 4,2 , or 0 Possible \# of imaginary zeros: 4,2 , or 0 d \# of complex zeros: 10 Possible \# of real zeros: 4,2 , or 0 Possible \# of imaginary zeros: 4,2 , or 0

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UpStudy AI · Accepted Answer

To determine the number of complex zeros and the possible number of real and imaginary zeros for the function $$ f(x) = x^4 - 9x^2 + 18 $$, we can use the following steps:

1. **Identify the degree of the polynomial**: The degree of the polynomial $$ f(x) $$ is 4, which means it has a total of 4 complex zeros (counting multiplicities) according to the Fundamental Theorem of Algebra.

2. **Determine the possible number of real zeros**: The number of real zeros can be found using Descartes' Rule of Signs, which states that the number of positive real zeros is equal to the number of sign changes in $$ f(x) $$ when evaluated at positive values, and the number of negative real zeros is equal to the number of sign changes in $$ f(-x) $$.

3. **Calculate $$ f(-x) $$**:
   $$
   f(-x) = (-x)^4 - 9(-x)^2 + 18 = x^4 - 9x^2 + 18
   $$
   Since $$ f(-x) = f(x) $$, there are no sign changes, indicating that there are no negative real zeros.

4. **Evaluate $$ f(x) $$ for positive real zeros**:
   - The function can be rewritten as:
   $$
   f(x) = (x^2 - 3)^2 + 9
   $$
   - This shows that $$ f(x) $$ is always positive for all real $$ x $$ since the square term is non-negative and the constant term is positive.

5. **Conclusion**:
   - Since $$ f(x) $$ has no real zeros, all 4 complex zeros must be non-real (imaginary).
   - Therefore, the number of complex zeros is 4, the possible number of real zeros is 0, and the possible number of imaginary zeros is 4.

### Final Answer:
- **Number of complex zeros**: 4
- **Possible number of real zeros**: 0
- **Possible number of imaginary zeros**: 4

Thus, the correct option is:
- a: \# of complex zeros: 4, Possible \# of real zeros: 0, Possible \# of imaginary zeros: 4.
The function $$ f(x) = x^4 - 9x^2 + 18 $$ has 4 complex zeros, all of which are imaginary.

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