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The graph in the image shows the polynomial function \( f(x) = x^6 - 7x^4 + 15x^2 - 9 \), and you are asked to determine the number of rational, irrational, and imaginary zeros of the function based on the Fundamental Theorem of Algebra and the graph provided.

### Key Information:

1. **Degree of the Polynomial:**
   The degree of the polynomial is 6 (from the \(x^6\) term), meaning the function can have up to 6 zeros (real or imaginary).

2. **Rational Zeros:**
   The graph appears to intersect the x-axis at least 2 points, meaning there are likely 2 real rational zeros. Since the function is even (only even powers of \(x\)), it is symmetric about the y-axis, which suggests symmetry in the real zeros as well.

3. **Irrational Zeros:**
   The remaining real zeros, if not rational, will be irrational. Based on the appearance of the graph, there seem to be 2 irrational zeros.

4. **Imaginary Zeros:**
   Since the total number of zeros (real and imaginary) must equal the degree of the polynomial (which is 6), and we can see at least 4 real zeros on the graph, the remaining zeros must be imaginary. This leaves us with 2 imaginary zeros.

### Conclusion:

- **Rational zeros**: 2
- **Irrational zeros**: 2
- **Imaginary zeros**: 2

Would you like more details on the reasoning behind these conclusions or explanations about how the Fundamental Theorem of Algebra applies here? Here are a few follow-up questions to expand the concepts:

1. How do we classify rational vs irrational zeros in a polynomial function?
2. How does the degree of the polynomial relate to the total number of zeros?
3. What is the symmetry in polynomial functions, and how does it affect the graph?
4. Can a polynomial have both real and complex conjugate pairs of zeros?
5. How can the multiplicity of zeros be determined from a graph?

**Tip:** The Fundamental Theorem of Algebra ensures that a polynomial of degree \(n\) will always have exactly \(n\) roots, counting multiplicity, whether real or imaginary.

The graph of the function f(x) = x^6 - 7x^4 + 15x^2 - 9 is shown. Use the graph and the Fundamental Theorem of Algebra to determine how many rational, irrational, and imaginary zeros there are for the function.

Learn how to use the Fundamental Theorem of Algebra to determine the number of rational, irrational, and imaginary zeros of the polynomial function f(x) = x^6 - 7x^4 + 15x^2 - 9. Suitable for high school students studying polynomial functions.

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