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.