The graph provided shows a degree 4 polynomial $f(x)$, and we are tasked with finding the factored form of the polynomial. Here's how we can proceed based on the visual information:

1. Roots:

We can identify the roots (x-intercepts) from the graph, which are the points where the polynomial crosses the x-axis:

• $x = -4$
• $x = 0$
• $x = 1$

2. Multiplicity:

We can observe the behavior of the graph at these roots to infer the multiplicities:

• At $x = -4$, the graph touches the x-axis but doesn't cross it, indicating a root with even multiplicity. Since the degree of the polynomial is 4, this root likely has multiplicity 2.
• At $x = 0$, the graph crosses the x-axis, indicating a root with odd multiplicity. This root likely has multiplicity 1.
• At $x = 1$, the graph again crosses the x-axis, suggesting another root with multiplicity 1.

3. Factored Form:

Given this information, the factored form of $f(x)$ can be written as: $f(x) = a(x + 4)^2 x (x - 1)$ where $a$ is a constant coefficient that can be determined by looking at other points on the graph, such as the y-intercept or a maximum/minimum point. From the graph, we can see that the polynomial has a local minimum around $y = -5$, which might help determine $a$.

Since no specific y-values are given beyond the intercepts, we leave $a$ as an arbitrary constant for now.

Summary:

The factored form is: $f(x) = a(x + 4)^2 x (x - 1)$ Would you like to explore how to determine the value of $a$, or any other details?

Here are 5 related questions:

1. How do we determine the constant $a$ from the graph?
2. What happens if a polynomial root has higher multiplicity?
3. How can we use the degree of a polynomial to predict the shape of its graph?
4. What is the significance of a root where the graph touches but does not cross the x-axis?
5. Can we use calculus (like the derivative) to verify the behavior of the graph at specific points?

Tip: Always check the multiplicities of roots by observing whether the graph crosses the x-axis (odd multiplicity) or just touches it (even multiplicity).