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Let's analyze the given graph of the degree 5 polynomial and fill in the table. We need to determine the x-values of the zeros (roots) and their multiplicities.

### Step 1: Identify the zeros
The graph crosses the x-axis at two points and touches it at one:
- The graph crosses the x-axis at \( x = -4 \) and \( x = 1 \).
- It touches the x-axis at \( x = 0 \), indicating a root with an even multiplicity (it doesn't cross but just touches the axis).

### Step 2: Determine the multiplicities
- The root at \( x = -4 \) has a multiplicity of 1 because the graph crosses the axis.
- The root at \( x = 1 \) also has a multiplicity of 1 because it crosses the axis.
- The root at \( x = 0 \) has a multiplicity of 2 because it touches the axis but doesn't cross it.

### Completed Table:
| Root with \( x = \) | Multiplicity |
|---------------------|--------------|
| -4                  | 1            |
| 1                   | 1            |
| 0                   | 2            |

Would you like further clarification on any step or more details on polynomial root behavior?

Here are 5 related questions to explore the concept further:
1. What is the significance of a root's multiplicity in a polynomial?
2. How does a root with even multiplicity affect the graph's behavior at that point?
3. Can you explain the relationship between the degree of a polynomial and the number of its real roots?
4. What happens to the graph of a polynomial at a root with odd multiplicity?
5. How would increasing the multiplicity of a root affect the graph?

**Tip:** The sum of the multiplicities of the roots equals the degree of the polynomial.

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

Explore how to determine the zeros and their multiplicities from the graph of a degree 5 polynomial. The solution includes identifying zeros at x = -4, 0, and 1, with detailed explanation on their multiplicity based on how the graph intersects or touches the x-axis. Suitable for high school students studying advanced algebra or pre-calculus.

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