https://file.aiquickdraw.com/mathbot/file/1726467415090holw2cfu.jpg

Let's analyze the graph and the provided polynomial options step by step.

### Observations from the Graph:
- The graph crosses the x-axis at \(x = -2\) and \(x = 3\), meaning these are the roots of the polynomial.
- At \(x = -2\), the graph touches the x-axis without crossing it. This suggests a **double root** at \(x = -2\).
- At \(x = 3\), the graph crosses the x-axis sharply, which is likely a **single root**.
- The polynomial seems to have a positive leading coefficient because it goes upwards on both ends (as \(x \to \infty\) and \(x \to -\infty\)).
  
### Analyzing the Options:
Given the behavior of the graph at the roots:
1. **Double root at \(x = -2\)** means the factor \((x + 2)\) should be squared.
2. **Single root at \(x = 3\)** means the factor \((x - 3)\) should not be squared.

Now, let's evaluate each option:

- **(A)**: \( f(x) = \frac{1}{20}(x + 2)(x - 3) \)
    - This option has a single root at \(x = -2\), but we know it should be a double root. So, this cannot be correct.

- **(B)**: \( f(x) = \frac{1}{20}(x + 2)^2(x + 3)^2 \)
    - This has a double root at \(x = -2\), which is good. However, it has another double root at \(x = 3\), which is incorrect. The graph shows a single root at \(x = 3\), so this option is incorrect.

- **(C)**: \( f(x) = -\frac{1}{20}(x + 2)^2(x - 3)^2 \)
    - This has the correct double root at \(x = -2\), but it has a double root at \(x = 3\), which should not be the case. Also, the negative sign would flip the graph, making it go downward on both ends, which does not match the graph. Thus, this is incorrect.

- **(D)**: \( f(x) = \frac{1}{20}(x + 2)^2(x - 3)^2(x^2 + 1) \)
    - This option has too many factors, including an extra quadratic factor \((x^2 + 1)\), which introduces additional complexity that is not present in the graph. Therefore, this is not correct.

### Conclusion:
None of the options perfectly match the behavior of the graph, but **Option B** comes the closest with the double root at \(x = -2\). However, it mistakenly gives a double root at \(x = 3\), so the actual correct form should have \( (x - 3) \) without squaring. Therefore, a potential correct option should be a modification of **B**: \(\frac{1}{20}(x + 2)^2(x - 3)\), which is not exactly given in the options.

Would you like more details or have any questions on this problem?

Here are five follow-up questions for deeper understanding:
1. What would happen to the graph if the leading coefficient were negative?
2. How does the multiplicity of a root affect the shape of the graph at that point?
3. Can you explain why \( (x - 3) \) corresponds to a single root based on the graph?
4. What does the factor \( x^2 + 1 \) represent, and why is it irrelevant here?
5. How do you determine the degree of the polynomial from the graph?

**Tip:** Remember, the behavior at the x-axis (whether the graph crosses or touches) gives key insights into the multiplicity of the roots!

The graph of the polynomial function f is shown above. Which of the following could be the equation for f?

Analyze the polynomial function and its graph to identify the equation that best matches the root behavior. Learn about the multiplicity of roots, how graphs behave at different types of roots, and how to identify polynomial equations from graphs. Suitable for Grades 11-12.

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