The problem asks you to write an equation for the 4th-degree polynomial based on the provided graph and factored form. The general structure given is:

$-k(x + 1)(x - 1)^2(x - 3)$

Key Points:

1. Roots (x-intercepts): The graph crosses the x-axis at $x = -1$, $x = 1$, and $x = 3$.

   • $x = -1$ is a single root, as the graph passes through the axis.
   • $x = 1$ is a double root, as the graph just touches the axis without crossing.
   • $x = 3$ is a single root.

2. Shape of the graph:

   • Since the graph opens downward and has two peaks, the leading coefficient must be negative. Therefore, $k$ is positive in the expression $-k$, ensuring the correct sign for the leading coefficient.

General form of the polynomial:

$y = -k(x + 1)(x - 1)^2(x - 3)$

We need to determine $k$ by using a point on the graph, but since no specific point other than the roots is given, you may either leave the expression with $k$ undetermined or estimate the value of $k$ based on a known point.

Do you have any specific point you want to use to calculate $k$, or would you like to proceed with this form?

Relative Questions:

1. How do you determine the multiplicity of roots from the graph?
2. What role does the leading coefficient play in determining the graph’s orientation?
3. What happens to the graph when you change the value of $k$?
4. How would you find the value of $k$ using a known point on the graph?
5. Why is it important to consider the behavior at each x-intercept (touching or crossing the axis)?

Tip: When analyzing polynomial graphs, always check the multiplicity of each root and the leading coefficient to get the correct shape of the graph.