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The graph provided shows a polynomial function, and the task is to write its formula for the polynomial function of least degree.

Here’s how we can approach it:

### Step 1: Identify the Zeros (Roots)
From the graph:
- The function crosses the x-axis at approximately \(x = -2\), \(x = 0\), and \(x = 3\). Therefore, the zeros of the function are \(x = -2\), \(x = 0\), and \(x = 3\).

### Step 2: Determine the Multiplicity of Each Root
- At \(x = -2\), the graph passes through the axis, indicating it has a root of multiplicity 1.
- At \(x = 0\), the graph also passes through the axis, which suggests this root also has a multiplicity of 1.
- At \(x = 3\), the graph touches the axis and bounces back up, which indicates a root with an even multiplicity. A multiplicity of 2 is the simplest assumption for the least degree.

### Step 3: Construct the Polynomial
Given the roots, the polynomial should include factors that correspond to each root:
- For \(x = -2\), the factor is \((x + 2)\).
- For \(x = 0\), the factor is \(x\).
- For \(x = 3\), the factor is \((x - 3)^2\).

Thus, the polynomial is of the form:
\[
f(x) = a \cdot x \cdot (x + 2) \cdot (x - 3)^2
\]
where \(a\) is a constant to be determined.

### Step 4: Estimate the Leading Coefficient
By looking at the behavior of the graph as \(x \to \infty\), the graph opens upwards, meaning \(a\) is positive. To further refine \(a\), we can use the y-intercept (which appears to be at around \(f(0) = 0\)) to determine the precise value of \(a\), but for now, assume \(a = 1\) for simplicity.

Thus, the polynomial can be written as:
\[
f(x) = x(x + 2)(x - 3)^2
\]

Would you like further details or explanations about any step?

Here are some additional questions to deepen understanding:
1. How does the multiplicity of a root affect the shape of the graph at that root?
2. What happens to the degree of a polynomial when a root has higher multiplicity?
3. How can you verify if the leading coefficient is correct using another point on the graph?
4. What are some methods for identifying the degree of a polynomial based on its graph?
5. How would the graph behave differently if \(a\) were negative?

**Tip:** Always double-check for symmetry in the graph when determining the form of the polynomial—this can give clues about the degree and behavior of the function.

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

Learn how to determine the polynomial function of least degree from a graph showing zeros at x = -2, 0, and 3. This example covers polynomial roots, root multiplicity, and constructing the corresponding polynomial function.

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