Looking at the graph you uploaded, the task is to find the expression for the polynomial in factored form that represents the curve with the least possible degree.

Steps to follow:

1. Identify the roots (x-intercepts): The graph crosses the x-axis at the following points:

   • $x = -4$
   • $x = -2$
   • $x = 1$
   • $x = 3$

   These points will correspond to factors of the polynomial.

2. Determine the multiplicity:

   • At $x = -4$ and $x = 1$, the graph crosses the x-axis, meaning these roots have odd multiplicity, likely 1.
   • At $x = -2$ and $x = 3$, the graph touches the x-axis but doesn't cross it, which means these roots have even multiplicity, likely 2.

3. General form: Based on the above information, the polynomial will have the following factored form: $y(x) = a(x + 4)(x + 2)^2(x - 1)(x - 3)^2$ where $a$ is a constant that determines the leading coefficient and the direction of the graph.

4. Determine the sign of $a$: The graph falls to negative infinity on the right side as $x$ increases, indicating that the leading term is negative. Thus, $a$ must be negative.

Final Answer:

$y(x) = -a(x + 4)(x + 2)^2(x - 1)(x - 3)^2$ where $a$ is likely to be $1$ based on a typical polynomial of minimal degree.

Do you want further details on how I chose the multiplicities or how to determine the leading coefficient?

Related Questions:

1. How do you determine the degree of a polynomial from its graph?
2. What is the role of multiplicity in determining the shape of a polynomial's graph?
3. How can you verify if a polynomial matches a given graph?
4. How do changes in the leading coefficient affect the graph of a polynomial?
5. Why do certain roots result in the graph only touching the x-axis instead of crossing it?

Tip:

When the graph touches the x-axis without crossing it, the root has even multiplicity. This behavior is useful in sketching polynomial functions.