https://file.aiquickdraw.com/mathbot/file/1726753503237iik1qzn8.jpg

To solve the question, we need to analyze the graph and determine the correct polynomial function \( p(x) \).

### Step-by-step Analysis:

1. **Roots/Zeros of the Polynomial**:
   The graph crosses the x-axis at \( x = -3 \) and \( x = 2 \), which means the polynomial must have factors of \( (x + 3) \) and \( (x - 2) \).

2. **Multiplicity of Roots**:
   - The graph **touches** the x-axis at \( x = -3 \) and **bounces off** at \( x = 2 \). This indicates that:
     - At \( x = -3 \), the root has **even** multiplicity (since it just touches the x-axis).
     - At \( x = 2 \), the root has **even** multiplicity as well (since the graph bounces off the x-axis).

3. **Degree of the Polynomial**:
   Based on the shape of the graph (with two turning points), the degree of the polynomial seems to be at least 4.

### Matching the Polynomial with the Given Options:
- From the analysis, the factors are likely \( (x + 3)^2 \) and \( (x - 2)^2 \) because both roots have even multiplicity.

Thus, the correct polynomial expression is:

\[
(x + 3)^2(x - 2)^2
\]

### Answer:
The correct option is the third one: \( (x + 3)^2(x - 2)^2 \).

Would you like further details or explanations on any part of this? Here are some related questions to explore:

1. What is the general form of a polynomial with given roots?
2. How do multiplicities of roots affect the graph of a polynomial?
3. What is the significance of turning points in determining the degree of a polynomial?
4. How can you factor a polynomial given its graph?
5. What are the characteristics of polynomial graphs with even and odd multiplicity?

**Tip:** Remember that an even multiplicity causes the graph to touch or bounce off the x-axis, while an odd multiplicity causes it to cross.

The graph of a polynomial function y = p(x) is shown. Which of the following expressions could define p(x)?

Learn how to identify the correct polynomial function based on its graph, using concepts like root multiplicity and polynomial degree. This problem is ideal for high school students in Grades 10-12.

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