https://file.aiquickdraw.com/mathbot/file/17300878704502ezxzq61.jpg

To analyze the graph and answer the question:

1. **Identify the zeros** of the polynomial by looking for points where the graph crosses or touches the x-axis.
2. **Determine the multiplicity** of each zero:
   - If the graph crosses the x-axis, the zero has an **odd multiplicity**.
   - If the graph only touches the x-axis and turns around, the zero has an **even multiplicity**.

### Step-by-Step Analysis:
Based on the graph:

- **Zero at \( x = -4 \):**
  - The graph **crosses the x-axis** at \( x = -4 \), so the multiplicity is likely **1**.

- **Zero at \( x = 0 \):**
  - The graph **touches** the x-axis and **turns around** at \( x = 0 \), indicating an **even multiplicity**. Given the maximum possible multiplicity is 3, this likely has **multiplicity 2**.

- **Zero at \( x = 5 \):**
  - The graph **crosses** the x-axis at \( x = 5 \), suggesting an **odd multiplicity**. Since it crosses only once, the likely multiplicity is **1**.

### Filling in the Table:
| Zeros | Multiplicity |
|-------|--------------|
| -4    | 1            |
| 0     | 2            |
| 5     | 1            |

Would you like further details or have any questions?

### Related Questions:
1. What are the rules to determine multiplicity based on graph behavior at each zero?
2. How would the graph look if the multiplicity at \( x = 0 \) were 3 instead of 2?
3. Can you identify if the function is of odd or even degree based on the end behavior?
4. How does the presence of zeros affect the general shape of the polynomial?
5. Why does an even multiplicity cause the graph to only touch the x-axis without crossing?

### Tip:
For polynomials, each zero's behavior can be predicted by its multiplicity: **even multiplicity** zeros "bounce" on the axis, while **odd multiplicity** zeros cross the axis.

Use the graph of the polynomial function shown below to identify the zeros of the function and the multiplicity of each zero. The largest possible multiplicity is three. Leave any remaining answer boxes empty.

Learn how to identify the zeros of a polynomial function and determine the multiplicity of each zero by analyzing the graph. This guide provides a step-by-step solution to understanding the behavior of polynomial functions based on their zeros and multiplicities. Ideal for students in Grades 10-12.

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