https://file.aiquickdraw.com/mathbot/file/1727322203644lgd0ce6i.jpg

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.

Use the graph to write the formula for a polynomial function of least degree.

Learn how to determine the polynomial function of least degree based on a graph with zeros at x = -3, x = 1, and x = 3. The solution explores key concepts like zeros, multiplicity, and factored form of polynomials. Suitable for high school precalculus students.

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