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?

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.