https://file.aiquickdraw.com/mathbot/file/1726363566047rmz7gzr2.jpg

The given graph suggests that the polynomial has the following key features:

1. **X-intercepts (roots)**: These are the points where the graph crosses or touches the x-axis. From the graph:
   - The graph crosses the x-axis at \( x = -3 \), \( x = -1 \), and \( x = 2 \).
   - At \( x = -1 \), the graph touches the x-axis without crossing it, which suggests a repeated root (multiplicity 2).

2. **Behavior at the intercepts**:
   - At \( x = -3 \), the graph crosses the x-axis, indicating that the root has multiplicity 1.
   - At \( x = -1 \), the graph touches and rebounds, indicating a root with multiplicity 2.
   - At \( x = 2 \), the graph crosses the x-axis, indicating multiplicity 1.

3. **End behavior**: The ends of the graph are moving in opposite directions, indicating that the polynomial is of odd degree, and the leading coefficient is positive.

Thus, the factored form of the polynomial can be written as:

\[
y(x) = a(x + 3)(x + 1)^2(x - 2)
\]

Now, to determine the value of \( a \), we need to use another point on the graph. From visual inspection, the y-intercept is approximately at \( y = 4 \) when \( x = 0 \). Substituting \( x = 0 \) and \( y = 4 \) into the equation:

\[
4 = a(0 + 3)(0 + 1)^2(0 - 2)
\]

\[
4 = a(3)(1)^2(-2)
\]

\[
4 = -6a
\]

Solving for \( a \):

\[
a = -\frac{2}{3}
\]

Thus, the polynomial in factored form is:

\[
y(x) = -\frac{2}{3}(x + 3)(x + 1)^2(x - 2)
\]

Would you like further details or clarification on any step?

Here are some related questions you can explore:

1. How does the multiplicity of a root affect the graph's behavior?
2. What does the degree of the polynomial indicate about the end behavior of the graph?
3. Why does the leading coefficient affect the direction of the graph?
4. How would the graph look if one of the roots had a higher multiplicity?
5. What steps can be used to verify the polynomial equation from a graph?

**Tip**: The number of turning points in a graph can help estimate the degree of the polynomial. A degree \( n \) polynomial can have up to \( n-1 \) turning points.

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

This lesson teaches how to factor a polynomial with given x-intercepts and multiplicity, based on its graph. It explains the behavior at roots -3, -1, and 2, and how to determine the polynomial's degree and leading coefficient from the graph's end behavior.

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