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To determine the polynomial function \( f(x) \) that matches the given graph, we need to:

1. Identify the roots of the polynomial from the graph, which correspond to the points where the graph crosses the x-axis.
2. Determine the degree of the polynomial based on the shape of the graph.
3. Use the leading coefficient, which is given as \( a = 1 \).

### Step 1: Identifying Roots
Looking at the x-axis intersections on the graph:
- The graph appears to cross or touch the x-axis at \( x = -5 \), \( x = 0 \), and \( x = 3 \).

### Step 2: Determining the Multiplicity of Each Root
- The root at \( x = -5 \) appears to cross the x-axis, indicating a **single root** (multiplicity 1).
- The root at \( x = 0 \) also crosses the x-axis, suggesting it is a **single root** as well.
- The root at \( x = 3 \) has a similar behavior, crossing the x-axis, so it is also a **single root**.

### Step 3: Writing the Polynomial in Factored Form
With each root having multiplicity 1 and \( a = 1 \), the polynomial function can be written as:
\[
f(x) = (x + 5)(x)(x - 3)
\]

### Step 4: Expanding if Necessary
If you need to expand this expression for further analysis, it would be:
\[
f(x) = (x + 5)(x)(x - 3) = x(x + 5)(x - 3) = x(x^2 + 2x - 15) = x^3 + 2x^2 - 15x
\]

So, the final polynomial in standard form is:
\[
f(x) = x^3 + 2x^2 - 15x
\]

Let me know if you want further details or have questions.

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Here are five related questions that build on this topic:

1. What is the significance of the multiplicity of roots in polynomial functions?
2. How would the polynomial change if one of the roots had a multiplicity of 2?
3. How does the leading coefficient affect the shape of the polynomial graph?
4. Can you determine the end behavior of the polynomial based on its degree?
5. How would we identify turning points of the polynomial from its equation?

**Tip:** The degree of a polynomial gives insight into the number of turning points and the end behavior of its graph.

Complete the f(x) function by writing a polynomial function in factored form that matches the graph. Use a = 1.

Learn how to write a polynomial function in factored form that matches a given graph. This guide covers identifying roots and their multiplicity, using a leading coefficient of 1, and constructing the polynomial function in factored form.

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