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To write an expression in factored form for the polynomial of least possible degree from the given graph, let's analyze the key features:

### Steps:
1. **Identify the x-intercepts (roots):**
   The graph crosses the x-axis at approximately \(x = -4\), \(x = 0\), and \(x = 3\). These are the roots of the polynomial.

2. **Determine the multiplicity of roots:**
   - At \(x = -4\) and \(x = 3\), the graph crosses the x-axis, indicating the roots have **multiplicity 1**.
   - At \(x = 0\), the graph touches the x-axis but does not cross it, which means this root has **even multiplicity**. Given that the graph seems to bounce off the x-axis smoothly, it suggests a multiplicity of **2**.

3. **Factor form:**
   A polynomial can be expressed in factored form using its roots and their multiplicities. For this graph:
   \[
   y(x) = a(x + 4)(x - 0)^2(x - 3)
   \]
   Simplifying:
   \[
   y(x) = a(x + 4)x^2(x - 3)
   \]
   Where \(a\) is a constant that represents the leading coefficient.

4. **Determine the sign of \(a\):**
   Since the graph opens downwards, the leading coefficient \(a\) must be negative.

### Final expression:
\[
y(x) = -1(x + 4)x^2(x - 3)
\]
Thus, the factored form of the polynomial is:
\[
y(x) = -(x + 4)x^2(x - 3)
\]

Let me know if you'd like further details or explanations! Here are five related questions:

1. How do you determine the multiplicity of a root by looking at a graph?
2. Why does the graph touch but not cross the x-axis at \(x = 0\)?
3. How would the graph look if the multiplicity at \(x = 0\) was 3 instead of 2?
4. What happens to the shape of the graph when the leading coefficient \(a\) is positive instead of negative?
5. Can a polynomial of higher degree fit this graph, and what would it look like?

**Tip:** Always check the direction of the graph (upwards or downwards) to determine the sign of the leading coefficient.

Write an expression in factored form for the polynomial of least possible degree graphed below.

Explore how to find the factored form of a polynomial from a graph. Learn about roots, multiplicity, and how the leading coefficient affects the graph. Suitable for students in Grades 9-12.

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