Let's analyze the given polynomial $f(x) = -3x^2(x + 7)(x - 5)^3$.

Step A: Identifying the x-intercepts and their multiplicities

To find the x-intercepts, we set $f(x) = 0$. The polynomial factors give the x-intercepts directly:

1. Factor 1: $x^2$ contributes a root at $x = 0$ with multiplicity 2.
2. Factor 2: $x + 7$ contributes a root at $x = -7$ with multiplicity 1.
3. Factor 3: $(x - 5)^3$ contributes a root at $x = 5$ with multiplicity 3.

The x-intercepts, in increasing order, are:

• Smallest zero: $x = -7$, multiplicity = 1
• Second zero: $x = 0$, multiplicity = 2
• Third zero: $x = 5$, multiplicity = 3

Step B: Behavior at each intercept

• Crosses the x-axis: If the multiplicity of the root is odd, the graph crosses the x-axis at that point.
• Touches the x-axis and turns: If the multiplicity of the root is even, the graph touches the x-axis and turns around.

From the multiplicities:

1. $x = -7$: Multiplicity 1 → Graph crosses the x-axis.
2. $x = 0$: Multiplicity 2 → Graph touches the x-axis and turns.
3. $x = 5$: Multiplicity 3 → Graph crosses the x-axis.

Final Answers

A) Identify the x-intercepts and their multiplicities:

• Smallest zero: $x = -7$, multiplicity = 1
• Second zero: $x = 0$, multiplicity = 2
• Third zero: $x = 5$, multiplicity = 3

B) Determine behavior:

• Zeros where the graph crosses the x-axis: $x = -7$, $x = 5$
• Zeros where the graph touches the x-axis and turns: $x = 0$

Would you like further details or visualizations for the graph?

Five related questions to explore further:

1. How does the negative leading coefficient affect the end behavior of the graph?
2. How would you find the y-intercept for the given polynomial?
3. What is the degree of the polynomial, and how does it affect the shape of the graph?
4. How do you find turning points for such a polynomial?
5. How would the graph change if the multiplicities of the factors were altered?

Tip: Always check the sign of the leading coefficient to determine how the ends of the graph behave.