To identify the correct equation of the polynomial $p(x)$ from the graph:

1. Identify the x-intercepts: The graph intersects the x-axis at $x = -1$, $x = -2.5$, and $x = 3$. These are the roots of the polynomial.

2. Check the behavior at each intercept:

   • At $x = -1$, the graph touches the x-axis and turns around, indicating a double root.
   • At $x = -2.5$, the graph crosses the x-axis, indicating a single root.
   • At $x = 3$, the graph touches the x-axis and turns around, indicating a double root.

3. Determine the corresponding factors:

   • $(x + 1)^2$ for the double root at $x = -1$.
   • $(2x + 5)$ for the single root at $x = -2.5$.
   • $(x - 3)^2$ for the double root at $x = 3$.

4. Choose the correct equation: The polynomial equation that fits these roots and their multiplicities is: $p(x) = (x + 1)^2 (2x + 5)(x - 3)^2$

This corresponds to option A in the provided choices.

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Related Questions:

1. How do you determine if a root is a double root or a single root from a graph?
2. What is the significance of the multiplicity of roots in polynomial graphs?
3. How does the leading coefficient affect the end behavior of the polynomial graph?
4. What happens when a polynomial graph has a triple root at a point?
5. Can a polynomial have complex roots, and how would they affect the graph?

Tip:

Always verify the behavior of the graph at each intercept to confirm the multiplicity of the roots when determining polynomial equations.