The problem is asking to identify the equation of the polynomial $p(x)$ based on its graph.

Steps to solve:

1. Identify the Zeros (Roots): The graph crosses or touches the x-axis at specific points. These are the zeros of the polynomial. From the image:

   • Zeros are at $x = -3$, $x = -1$, and $x = 2$.

2. Determine the Multiplicity:

   • At $x = -3$: The graph crosses the x-axis, so the zero has a multiplicity of 1.
   • At $x = -1$: The graph bounces off the x-axis, indicating a multiplicity of 2.
   • At $x = 2$: The graph crosses the x-axis, so the zero has a multiplicity of 1.

3. Write the Polynomial: Using the zeros and their multiplicities, the polynomial can be written as: $p(x) = a(x + 3)(x + 1)^2(x - 2)$ where $a$ is a leading coefficient to be determined based on additional information (like the y-intercept or behavior as $x \to \infty$).

4. Check the End Behavior: Since the polynomial appears to rise to infinity as $x \to \infty$, the leading coefficient $a > 0$. Hence, $a$ is positive.

Final Equation:

The equation is likely: $p(x) = a(x + 3)(x + 1)^2(x - 2)$ where $a$ is a positive constant.

Would you like me to help verify this further or explain any concepts? Let me know!

Related Questions:

1. What is the difference between a zero with multiplicity 1 and multiplicity 2?
2. How do you find the degree of a polynomial based on its graph?
3. How does the leading coefficient affect the graph's end behavior?
4. What is the importance of factoring a polynomial to find its zeros?
5. How do turning points relate to the degree of a polynomial?

Tip:

Always check the graph's symmetry and end behavior to determine whether the polynomial has an even or odd degree and a positive or negative leading coefficient!