https://file.aiquickdraw.com/mathbot/file/17260724974228l671wrh.jpg

The image shows the graph of a quartic polynomial function with zeros at \( A \), \( B \), and \( C \). The question asks to determine the multiplicity of the zero at \( B \) with the correct reasoning.

### Explanation:
- **Multiplicity** refers to how many times a particular root is repeated. If the graph touches the x-axis and turns around (like at point \( B \) in this graph), it indicates the zero has an **even multiplicity**. Specifically, if the graph only touches the x-axis at a point but does not cross it, the zero has a **multiplicity of 2**.
- If the graph crosses the x-axis, the multiplicity would be odd. Since \( B \) represents a point where the graph touches but does not cross, the multiplicity must be even, and in this case, it's likely **2**.

### Correct Answer:
The correct reasoning based on the graph is the fourth option:
- **2, because the graph of the function intersects the x-axis twice at \( B \)**.

Would you like further details on this explanation or have any other questions?

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Here are 5 related questions to further your understanding:
1. What is the general shape of a quartic polynomial?
2. How do you determine the degree of a polynomial by looking at its graph?
3. What does a zero with odd multiplicity indicate about the behavior of the graph?
4. How can you tell if a function is even or odd by looking at its graph?
5. What is the significance of the leading coefficient in determining the end behavior of a polynomial graph?

**Tip:** To identify multiplicity, always look at how the graph behaves at the zero: does it touch or cross the x-axis?

The figure shows the graph of a quartic polynomial function with zeros at A, B, and C. Which of the following statements gives the multiplicity of the zero at B with the correct reasoning?

Learn how to determine the multiplicity of the zero at point B in a quartic polynomial function by analyzing its graph. This problem covers polynomial concepts and the behavior of zeros with even or odd multiplicity. Suitable for high school students.

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