https://file.aiquickdraw.com/mathbot/file/1726252793778pl0m7spw.jpg

The polynomial function provided in the image is:

\[
P(x) = 3x(x + 1)^2(x - a)
\]

where \(a\) is a real number. We are asked to determine the graph of \(y = P(x)\).

### Key Steps to Analyze the Polynomial:

1. **Degree of the Polynomial**:
   The degree of the polynomial can be found by expanding \(P(x)\). The highest degree term comes from multiplying \(3x(x + 1)^2(x - a)\), which will result in a degree of 4, as we have the following factors contributing to the degree:
   - \(x\) is degree 1,
   - \((x + 1)^2\) is degree 2,
   - \((x - a)\) is degree 1.
   
   So, the polynomial is degree 4.

2. **Roots and Multiplicities**:
   The roots of the polynomial come from solving \(P(x) = 0\). The roots are:
   - \(x = 0\) (with multiplicity 1, as the factor is \(x\)),
   - \(x = -1\) (with multiplicity 2, as the factor is \((x + 1)^2\)),
   - \(x = a\) (with multiplicity 1, as the factor is \((x - a)\)).

3. **Behavior Near the Roots**:
   - At \(x = 0\), the polynomial crosses the x-axis because the root has multiplicity 1.
   - At \(x = -1\), the graph touches the x-axis but doesn't cross it, since the root has multiplicity 2 (even multiplicity implies a touch, not a cross).
   - At \(x = a\), the graph crosses the x-axis because this root also has multiplicity 1.

4. **End Behavior**:
   Since the polynomial is degree 4 and the leading coefficient (after expansion) is positive, the end behavior will be:
   - As \(x \to -\infty\), \(P(x) \to +\infty\),
   - As \(x \to +\infty\), \(P(x) \to +\infty\).

### Conclusion:
The graph of \(y = P(x)\) will have the following features:
- A crossing at \(x = 0\) and \(x = a\),
- A touch (not crossing) at \(x = -1\),
- Both ends of the graph rise to infinity.

Does this help, or would you like further clarifications on any part? 

### Related Questions:
1. How would the graph change if the coefficient in front of \(P(x)\) were negative?
2. What would happen to the graph if the root at \(x = -1\) had multiplicity 1 instead of 2?
3. How can the value of \(a\) impact the location of the graph's crossing point?
4. What is the effect of multiplicity on the behavior of polynomial graphs at the x-axis?
5. How can you determine the turning points of a polynomial graph based on its degree?

### Tip:
Always check the end behavior of a polynomial by examining its leading term; it helps predict the graph's shape at extreme values of \(x\).

The polynomial function P(x) is given by P(x) = 3x(x + 1)^2(x - a), where a is a real number. Which of the following could be the graph of y = P(x)?

Explore how to analyze the graph of the polynomial function P(x) = 3x(x + 1)^2(x - a). This problem covers key algebraic concepts such as root multiplicities, polynomial degree, and end behavior, suitable for students in Grades 10-12.

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