https://file.aiquickdraw.com/mathbot/file/1727179320555i5ycv502.jpg

The graph shown in the image is of a fourth-degree polynomial function. To answer the question regarding which option correctly shows the three factors of the function, we need to examine the behavior of the graph.

### Key Observations from the Graph:
1. **Roots of the Polynomial**: The graph crosses the x-axis at three distinct points: \(x = -5\), \(x = -2\), and \(x = 3\). These represent the **roots** or **zeros** of the function.
   
2. **Multiplicities**:
   - At \(x = -5\), the graph crosses the x-axis, implying a **multiplicity of 1**.
   - At \(x = -2\), the graph touches the x-axis and bounces back, implying a **multiplicity of 2**.
   - At \(x = 3\), the graph crosses the x-axis, implying a **multiplicity of 1**.

### Factors of the Polynomial:
Based on the observed roots and their multiplicities, the factors of the polynomial should be:
- \( (x + 5) \) for the root \(x = -5\),
- \( (x + 2)^2 \) for the root \(x = -2\) with multiplicity 2,
- \( (x - 3) \) for the root \(x = 3\).

Thus, the correct expression for the polynomial should include these factors.

### Evaluating the Answer Choices:
- **A) \( (x - 3)(x + 2)(x + 5) \)**: This is incorrect because it does not account for the multiplicity of 2 for \(x = -2\).
- **B) \( (x - 3)(x + 5) \)**: This is incorrect because it lacks the factor for \(x = -2\).
- **C) \( (x + 3)(x - 2)(x - 5) \)**: This is incorrect as the signs of the roots are wrong.
- **D) \( (x + 2)(x - 5)(x - 3) \)**: This is incorrect as it does not match the observed factors and their multiplicities.

Thus, none of the provided options match the correct factors for the polynomial based on the graph. If we had an option that represented \( (x - 3)(x + 2)^2(x + 5) \), that would be the correct answer.

The graph is that of a fourth-degree polynomial function. Which of the following correctly shows three factors of the function?

Learn how to identify the correct factors of a fourth-degree polynomial function based on its graph. Explore the concept of roots, multiplicities, and polynomial behavior. Ideal for students in Grades 9-12.

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