https://file.aiquickdraw.com/mathbot/file/1728250315515bou54s45.jpg

The graph you have provided represents a 5th-degree polynomial function. Based on the **Fundamental Theorem of Algebra**, we know that a polynomial of degree \(n\) has exactly \(n\) complex roots (including real and imaginary zeros).

### Analyzing the graph:
- **Real zeros** are where the graph crosses the x-axis. From the graph, we can see that the polynomial crosses the x-axis **once**.
  - Therefore, there is **1 real zero**.

- The total number of zeros (real and imaginary) must add up to 5 since it's a 5th-degree polynomial.
  - Since there is 1 real zero, the remaining \(5 - 1 = 4\) zeros must be imaginary.

### Conclusion:
- **Number of real zeros**: 1
- **Number of imaginary zeros**: 4

Would you like a more detailed explanation of this concept? Also, here are some related questions to explore further:

1. What is the Fundamental Theorem of Algebra?
2. How do we determine if a polynomial has imaginary roots by looking at the graph?
3. Why do imaginary zeros always come in pairs for polynomials with real coefficients?
4. Can a polynomial of odd degree have only imaginary roots?
5. How can we factor a polynomial to identify its real and imaginary zeros?

**Tip**: Remember, imaginary roots always occur in conjugate pairs, so the number of imaginary zeros in a polynomial with real coefficients will always be an even number!

The graph of a 5th-degree polynomial is shown. Use the Fundamental Theorem of Algebra to determine the number of real and imaginary zeros.

Explore how to determine the number of real and imaginary zeros of a 5th-degree polynomial using the graph and the Fundamental Theorem of Algebra. Suitable for high school students learning about polynomials and complex numbers.

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