The image shows a question about how the multiplicity of a zero determines the behavior of the graph of a polynomial function at that zero.

Problem Breakdown:

The problem involves a seventh-degree polynomial function with zeros at $x = -6$ (multiplicity 2), $x = 0$ (multiplicity 3), $x = 1$ (multiplicity 1), and $x = 4$ (multiplicity 1).

Multiplicity and Graph Behavior:

• Even Multiplicity (e.g., 2, 4, etc.): The graph touches the x-axis at the zero and turns around. It does not cross the x-axis.
• Odd Multiplicity (e.g., 1, 3, etc.): The graph crosses the x-axis at the zero.

Answer the Questions:

1. The graph of the function $\text{(Choose...)}$ at the zero $x = -6$:

   • Answer: touches the x-axis and turns around.
   • Explanation: The zero at $x = -6$ has a multiplicity of 2 (even), so the graph touches and turns around.

2. The graph of the function $\text{(Choose...)}$ at the zero $x = 0$:

   • Answer: crosses the x-axis.
   • Explanation: The zero at $x = 0$ has a multiplicity of 3 (odd), so the graph crosses the x-axis.

3. The graph of the function $\text{(Choose...)}$ at the zero $x = 1$:

   • Answer: crosses the x-axis.
   • Explanation: The zero at $x = 1$ has a multiplicity of 1 (odd), so the graph crosses the x-axis.

4. The graph of the function $\text{(Choose...)}$ at the zero $x = 4$:

   • Answer: crosses the x-axis.
   • Explanation: The zero at $x = 4$ also has a multiplicity of 1 (odd), so the graph crosses the x-axis.

Would you like more details on any part of this solution, or do you have any other questions?

Related Questions:

1. What happens if a zero has a multiplicity of 4? How does the graph behave at that zero?
2. How does the degree of a polynomial relate to the number of zeros it can have?
3. Can a polynomial have a zero with multiplicity 0? Why or why not?
4. What is the relationship between the sign of the leading coefficient and the end behavior of the graph?
5. How can you determine the degree of a polynomial by looking at its graph?

Tip:

When analyzing the behavior of polynomials, always consider both the multiplicity of the zeros and the leading coefficient to understand the overall shape and direction of the graph.