To solve this polynomial function problem based on the graph, we need to follow several steps to identify the polynomial equation:

Step 1: Identify the Zeros (x-intercepts)

From the graph, it looks like the function crosses the x-axis at the following points:

• $x = -3$
• $x = 1$
• $x = 3$

These are the zeros of the polynomial function.

Step 2: Determine the Multiplicity of Each Zero

Multiplicity refers to how the graph behaves at each zero:

• At $x = -3$, the graph crosses the x-axis, so the zero has multiplicity 1.
• At $x = 1$, the graph touches the x-axis and turns around, indicating multiplicity 2.
• At $x = 3$, the graph crosses the x-axis, meaning multiplicity 1.

Step 3: Write the Polynomial in Factored Form

Using the identified zeros and their multiplicities, we can write the polynomial in factored form: $f(x) = a(x + 3)(x - 1)^2(x - 3)$ where $a$ is a constant that scales the function. To find the value of $a$, we could use another point on the graph, but that information is not visible here, so we'll assume $a = 1$ unless we have another point.

Step 4: Final Answer

The polynomial function with the least degree is: $f(x) = (x + 3)(x - 1)^2(x - 3)$

Would you like further clarification on any part of this? Also, here are five related questions that could deepen your understanding:

1. How does the multiplicity of a root affect the shape of the graph at that root?
2. What would happen if the graph crossed the x-axis at $x = 1$ instead of touching it?
3. How do we determine the leading coefficient $a$ if more information is given about another point on the graph?
4. Can the degree of the polynomial change if a zero has a different multiplicity?
5. How would you find the turning points of this polynomial function?

Tip: Always check the behavior of the graph at each zero to determine the correct multiplicity, as it significantly affects the form of the polynomial equation.