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The image shows a problem asking which of the given polynomials could define the function \( P(x) \) represented by the graph.

### Key observations from the graph:
1. **Roots (x-intercepts):** The graph crosses the x-axis at \( x = -3 \), \( x = 0 \), and \( x = 2 \).
2. **Behavior near the roots:**
   - At \( x = 0 \), the graph touches the x-axis and turns back up, indicating a **repeated root** or a **double root**.
   - At \( x = -3 \) and \( x = 2 \), the graph crosses the x-axis, indicating **simple roots** at these points.

### Forming the polynomial based on this behavior:
- Since the graph touches the x-axis at \( x = 0 \), we have a **double root** at \( x = 0 \), meaning a factor of \( x^2 \).
- The simple roots at \( x = -3 \) and \( x = 2 \) correspond to factors \( (x + 3) \) and \( (x - 2) \), respectively.

Thus, the polynomial should take the form:
\[
P(x) = \frac{(x + 3)(x^2)(x - 2)}{12}
\]

### Checking the options:
- **Option B** matches this form:
  \[
  P(x) = \frac{(x + 3)(x^2)(x - 2)}{12}
  \]

Thus, the correct answer is **B**.

Would you like a deeper explanation of any step?

Here are five related questions:
1. How do you identify simple and repeated roots from a graph?
2. What is the effect of a double root on the shape of the graph?
3. How would the polynomial change if the root at \( x = 2 \) were a double root?
4. How does factoring a polynomial help in graphing it?
5. How does the leading coefficient affect the end behavior of a polynomial?

**Tip:** When analyzing graphs, always look at how the curve behaves near the x-intercepts to determine the multiplicity of the roots.

The graph of y = P(x) is shown. Which of the following could define P(x)?

Learn how to identify polynomial functions based on their graphs by analyzing roots and their multiplicities. This problem covers polynomial functions, graphing techniques, and root behavior, suitable for students in Grades 10-12.

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