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 $x$.