The problem asks you to determine which of the given polynomials could define the function $P(x)$, based on the graph shown.

Key Features of the Graph:

1. The function crosses the x-axis at two points: $x = -3$ and $x = 4$.
2. The behavior at $x = -3$ appears to be parabolic (indicating a double root at $x = -3$), and the function crosses the x-axis at $x = 4$ (indicating a single root).
3. The polynomial appears to be cubic since it has two turning points, which suggests the highest degree term should be $x^3$.

Examining the Options:

1. Option A: $P(x) = \frac{(x - 4)(x + 3)^2}{12}$

   • The factor $(x + 3)^2$ indicates a double root at $x = -3$, which matches the behavior at $x = -3$ on the graph.
   • The factor $(x - 4)$ indicates a single root at $x = 4$, which also matches the graph.
   • Therefore, this option seems like a good candidate.

2. Option B: $P(x) = \frac{(x - 4)^2(x + 3)}{12}$

   • This option has $(x - 4)^2$, indicating a double root at $x = 4$, which does not match the graph's behavior (the graph crosses the x-axis at $x = 4$, so it should be a single root).
   • $(x + 3)$ would indicate a single root at $x = -3$, which also does not match the graph.

Conclusion:

The correct answer is Option A, because it matches the root structure and the behavior at the intercepts shown in the graph.

Would you like more details on any part of this explanation?

Here are 5 related questions to deepen your understanding:

1. How does the degree of a polynomial relate to the number of turning points?
2. What is the difference between a single root and a double root in a graph?
3. Why does the factor $(x - a)^2$ represent a double root?
4. How can you determine the end behavior of a cubic polynomial?
5. How do scaling factors (like $\frac{1}{12}$) affect the shape of a graph?

Tip: Always check the behavior at the intercepts to determine if a root is single, double, or higher multiplicity.