https://file.aiquickdraw.com/mathbot/file/1730097002696f7qzfgg5.jpg

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

### Step 1: Identify the Roots and Their Multiplicities
The graph intersects or touches the x-axis at three points, which represent the roots of the polynomial. These roots and their behavior (how the graph crosses or touches the x-axis) help us determine the multiplicities:

1. **Root at \( x = -3 \):** The graph crosses the x-axis, indicating a root with **odd multiplicity**. We'll assume a multiplicity of 1.
2. **Root at \( x = -1 \):** The graph touches the x-axis and turns back, indicating a root with **even multiplicity**. We'll assume a multiplicity of 2.
3. **Root at \( x = 2 \):** The graph crosses the x-axis, indicating a root with **odd multiplicity**. We'll assume a multiplicity of 1.

### Step 2: Write the Polynomial in Factored Form
Using these roots and their assumed multiplicities, the polynomial can be expressed as:
\[
y(x) = a(x + 3)(x + 1)^2(x - 2)
\]
where \( a \) is a constant to be determined.

### Step 3: Determine the Leading Coefficient \( a \)
To determine \( a \), we need a specific point on the graph. If the graph shows a point where \( y(x) \) has a known value (such as \( y(0) \)), we could plug that in to solve for \( a \). However, if no such point is provided, we can leave \( a \) as a variable or assume \( a = 1 \) for simplicity.

If we assume \( a = 1 \), the polynomial becomes:
\[
y(x) = (x + 3)(x + 1)^2(x - 2)
\]

Would you like further details on any step, or do you have additional questions?

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### Related Questions to Expand Understanding:
1. How do you determine the degree of a polynomial based on its roots and multiplicities?
2. What does the multiplicity of a root tell us about the behavior of the graph at that root?
3. How can we determine the leading coefficient \( a \) if more information is provided?
4. How would the polynomial change if one of the roots had a different multiplicity?
5. Why do we assume the least possible degree for the polynomial in this problem?

### Tip:
When graphing polynomials, remember that the behavior of the graph at each root (crossing or touching the axis) indicates whether the root’s multiplicity is odd or even.

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

Learn how to write the expression in factored form for a polynomial of least possible degree by analyzing the graph. Understand key algebraic concepts like roots, multiplicities, and polynomial behavior at each root. Suitable for students in Grades 10-12.

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