Do you find any point or zero which has a multiplicity in the graph? If so, specify them with multiplicity and explain the reason

In the given graph, let's examine the zeros or x-intercepts to identify any multiplicity:

1. **Zero at \( D = (-2, 0) \):**
   - The graph touches the x-axis and then rebounds. This suggests a zero with even multiplicity. Given the behavior of the graph at this point (a turning point where the graph doesn't cross the x-axis), it is likely to be a zero of multiplicity 2.

2. **Zero at \( E = (-1, 0) \):**
   - The graph crosses the x-axis, indicating a zero with odd multiplicity. Since it appears to just cross the axis without any flattening or turning around, it likely has a multiplicity of 1.

3. **Zero at \( C = (2, 0) \):**
   - Similar to point \( E \), the graph crosses the x-axis, suggesting a zero with odd multiplicity. Again, it likely has a multiplicity of 1, as there is no indication of the graph flattening or turning around.

### Summary of Multiplicities:
- \( D = (-2, 0) \) has a **multiplicity of 2**.
- \( E = (-1, 0) \) has a **multiplicity of 1**.
- \( C = (2, 0) \) has a **multiplicity of 1**.

### Explanation:
- A zero with **even multiplicity** (like \( D = (-2, 0) \)) will touch the x-axis and bounce back without crossing it. This is characteristic of a zero where the graph changes direction at the x-axis.
- A zero with **odd multiplicity** (like \( E = (-1, 0) \) and \( C = (2, 0) \)) will cross the x-axis. This means the function changes sign at this point.

Would you like any further details or explanations?

Here are 5 questions you can ask next:
1. How does the multiplicity of a zero affect the shape of the graph at that point?
2. What are the implications of having a zero with even multiplicity in a polynomial function?
3. How can we determine the exact multiplicity of a zero algebraically?
4. What is the significance of the leading coefficient in determining the end behavior of a polynomial function?
5. How does the Fundamental Theorem of Algebra relate to the number of zeros of a polynomial function?

**Tip:** To confirm the multiplicity of a zero, look at the behavior of the graph near the zero and the corresponding factor in the polynomial equation.

Learn how to identify the multiplicity of zeroes in graphs of polynomial functions. Explore the characteristics of zeroes with even and odd multiplicities through detailed examples and explanations.

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