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Based on the provided graph of a degree 5 polynomial, we need to determine the x-values of the roots and their multiplicities.

### Analyzing the graph:
1. **Root at x = -4**: 
   - This root crosses the x-axis at \( x = -4 \) and the behavior suggests it has **multiplicity 1**. This is because the graph crosses the x-axis sharply.
   
2. **Root at x = 2**:
   - The graph touches the x-axis at \( x = 2 \) without crossing it, indicating this is a root with **multiplicity 2**. 

3. **Other roots**: 
   - The graph crosses the x-axis again near \( x = 0 \), which suggests a root at \( x = 0 \) with **multiplicity 1**.

### Updated table:

| Root with x = | Multiplicity |
|---------------|--------------|
| -4            | 1            |
| 0             | 1            |
| 2             | 2            |

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the general shape of a degree 5 polynomial based on its behavior at infinity?
2. How does multiplicity affect the graph of a polynomial at its roots?
3. What is the difference between even and odd multiplicity in terms of graph behavior?
4. Can you identify other key features of polynomials based on their graphs?
5. How do the number of roots and their multiplicities relate to the degree of the polynomial?

**Tip:** Always check how the graph behaves at the x-axis to determine the root's multiplicity—crossing means odd multiplicity, touching means even.

Given the graph of a degree 5 polynomial, complete the table of values for either the x-value of a zero, or the multiplicity of the zero.

This problem involves analyzing the graph of a degree 5 polynomial to determine the x-values of roots and their multiplicities. Students will learn how graph behavior reveals multiplicities—crossing the x-axis indicates odd multiplicity, while touching the axis suggests even multiplicity. Suitable for students in Grades 9-12.

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