https://file.aiquickdraw.com/mathbot/file/1726457656417py9h5ps8.jpg

From the image, you are given the graph of a degree 5 polynomial, and a table where you need to fill in the x-values of the zeros or the multiplicity of those zeros.

### Step 1: Identify the zeros of the polynomial.
The zeros of a polynomial are the x-values where the graph crosses or touches the x-axis. From the graph, the x-axis intersections happen at:
- \( x = -4 \)
- \( x = -1 \)
- \( x = 2 \)
- \( x = 4 \)

### Step 2: Determine the multiplicity of each zero.
- A zero has **odd multiplicity** if the graph **crosses** the x-axis at that point.
- A zero has **even multiplicity** if the graph **touches** the x-axis and then turns around without crossing.

#### Analyze the graph at each zero:
1. **At \( x = -4 \)**:
   - The graph **touches** the x-axis at \( x = -4 \) and turns around, which means the multiplicity is **even**. Since you are told that \( x = -4 \) has an even multiplicity and there is only one even possibility for this polynomial, its multiplicity is **2**.
   
2. **At \( x = -1 \)**:
   - The graph **crosses** the x-axis at \( x = -1 \), meaning it has an **odd** multiplicity. The multiplicity is **1**.

3. **At \( x = 2 \)**:
   - You are given that the multiplicity is **2**, which is even. The graph touches the x-axis at this point and turns around.

4. **At \( x = 4 \)**:
   - The graph **crosses** the x-axis at \( x = 4 \), so it has an **odd** multiplicity. The multiplicity is **1**.

### Step 3: Complete the table.

| Root with \( x = \) | Multiplicity |
|---------------------|--------------|
| \( -4 \)            | \( 2 \)      |
| \( -1 \)            | \( 1 \)      |
| \( 2 \)             | \( 2 \)      |
| \( 4 \)             | \( 1 \)      |

Let me know if you'd like further details or have any other questions!

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**Here are 5 follow-up questions you might find helpful**:
1. How is the degree of a polynomial related to the number of roots?
2. What is the significance of the multiplicity of a zero?
3. How do you determine if a zero has an odd or even multiplicity from a graph?
4. How would the graph change if one of the zeros had a higher multiplicity?
5. What is the maximum number of turning points for a polynomial of degree 5?

**Tip:** Always check whether the graph crosses or just touches the x-axis to determine the multiplicity of a zero.

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.

Explore how to identify the roots and their multiplicities from the graph of a degree 5 polynomial. Learn key polynomial concepts and determine whether the graph crosses or touches the x-axis at each zero.

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