https://file.aiquickdraw.com/mathbot/file/1725925725446jp9rsrcx.jpg

The graph shows a degree 5 polynomial, and you're asked to complete the table of values based on the x-values of the zeros and their corresponding multiplicities.

### Analyzing the graph:

1. The polynomial crosses the x-axis at \(x = -4\), indicating a zero at \(x = -4\).
   - Since the graph crosses the x-axis at this point, the multiplicity of this root is **odd**. We already see in the table that its multiplicity is missing.
   
2. The polynomial also crosses the x-axis at \(x = 1\), and since it crosses straight through, the multiplicity is **1** (this value is already given in the table).

3. The graph touches the x-axis at approximately \(x = -1\), meaning it doesn't cross but touches the x-axis and turns around, indicating a **multiplicity greater than 1** (typically 2).

### Table Completion:

- The zero at \(x = -4\) has **multiplicity 1** because it crosses the x-axis.
- The root at \(x = -1\) should be added to the table with **multiplicity 2**.

Here's the completed table:

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

Would you like a more detailed explanation of how multiplicities work, or have any questions?

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Here are 5 related questions:
1. How can you tell if a root has an odd or even multiplicity just by looking at the graph?
2. What would happen if a root has a multiplicity of 3?
3. Can a polynomial of degree 5 have more than 5 roots?
4. How does the degree of the polynomial affect the shape of the graph near a root with multiplicity greater than 1?
5. How can you factor this polynomial based on the roots identified?

**Tip:** The degree of a polynomial dictates the maximum number of roots (real or complex) it can have.

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.

Analyze a degree 5 polynomial from a graph to determine its zeros and corresponding multiplicities. Learn how crossing and touching points on the x-axis represent odd and even multiplicities. Suitable for Grades 10-12 students learning advanced algebra and polynomial functions.

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