https://file.aiquickdraw.com/mathbot/file/1726236513603hrjnveoa.jpg

Let's analyze the given polynomial graph and equation options.

### Step 1: Identify the roots (x-intercepts)
From the graph:
- The polynomial has x-intercepts at \(x = -2\) and \(x = 2.5\).
- There's a "bounce" at \(x = 2.5\), indicating a double root.

Thus, the factors associated with these roots should be:
- For \(x = -2\), the factor is \((x + 2)\).
- For \(x = 2.5\) or \(x = 5/2\), the factor is \((2x - 5)\), and since this is a double root, it should be squared: \((2x - 5)^2\).

### Step 2: Analyze the options
Now, look at each option to see which matches the structure derived from the graph:
- Option **a**: \( p(x) = x^2(2x - 5)(x + 2) \)
  - This does not match because \( (2x - 5) \) is not squared, but the graph shows a double root at \(x = 2.5\).
  
- Option **b**: \( p(x) = x^2(2x - 5)(x + 2)^2 \)
  - This suggests a double root at \(x = -2\), but the graph shows a single root at \(x = -2\), so this is incorrect.
  
- Option **c**: \( p(x) = x^2(2x - 5)^2(x + 2) \)
  - This fits the structure of the graph: single root at \(x = -2\), double root at \(x = 2.5\), and matches the squaring of \( (2x - 5) \).
  
- Option **d**: \( p(x) = x(2x - 5)^2(x + 2) \)
  - This is close, but there is no additional \(x^2\) factor for proper degree.

### Step 3: Conclusion
The correct equation that matches the graph is **option c**, \( p(x) = x^2(2x - 5)^2(x + 2) \).

Would you like more details on the steps?

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Here are five related questions to expand on this:
1. What does it mean when a root "bounces" on the x-axis in a polynomial graph?
2. How do we determine the degree of a polynomial from its graph?
3. What is the significance of multiplicity of roots in polynomials?
4. How do you expand a polynomial like \( (2x - 5)^2(x + 2) \)?
5. Can a polynomial have both real and imaginary roots?

**Tip:** When a root touches the x-axis but does not cross, it has even multiplicity, typically 2 for a bounce effect.

Given a graph of polynomial function p, what could be the equation of the polynomial?

Learn how to determine the equation of a polynomial from its graph by analyzing the roots and their multiplicity. This example covers polynomial functions, including identifying double roots and factorization. Suitable for students in Grades 10-12.

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