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Let's solve the given problem step by step for the polynomial function \( f(x) = (x - 7)^3 (x + 9)^2 \).

### Part (a): List each real zero and its multiplicity.
- The zeros of the function are found by setting each factor to zero:
  - \( x - 7 = 0 \) gives \( x = 7 \), with a multiplicity of 3.
  - \( x + 9 = 0 \) gives \( x = -9 \), with a multiplicity of 2.

### Part (b): Determine whether the graph crosses or touches the x-axis at each x-intercept.
- At \( x = 7 \), the zero has an odd multiplicity (3), which means the graph will **cross** the x-axis.
- At \( x = -9 \), the zero has an even multiplicity (2), which means the graph will **touch** the x-axis but not cross it.

### Part (c): Determine the maximum number of turning points on the graph.
- The degree of the polynomial can be found by adding the multiplicities: \( 3 + 2 = 5 \).
- A polynomial of degree \( n \) can have at most \( n - 1 \) turning points. Therefore, the maximum number of turning points is \( 5 - 1 = 4 \).

### Part (d): Determine the end behavior (find the power function that the graph of \( f \) resembles for large values of \( |x| \)).
- For large values of \( |x| \), the term that dominates \( f(x) \) is \( (x - 7)^3 (x + 9)^2 \approx x^3 \cdot x^2 = x^5 \).
- Thus, the end behavior of the polynomial will resemble the power function \( f(x) \approx x^5 \).
- Since the leading term has an odd degree (5) with a positive coefficient, the graph will fall to negative infinity as \( x \to -\infty \) and rise to positive infinity as \( x \to +\infty \).

Would you like more details on any part of this explanation? Here are some related questions:

1. What is the effect of multiplicity on the shape of the graph at a zero?
2. How do you find the turning points of a polynomial function?
3. Why does the degree of the polynomial affect the end behavior?
4. Can a polynomial have more turning points than its degree minus one?
5. What happens to the graph if a factor is raised to a higher power?

**Tip:** When analyzing polynomial functions, pay attention to the multiplicity of each root to predict the graph's behavior at the intercepts.

For the polynomial function f(x) = (x - 7)^3 (x + 9)^2, answer the following questions:
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the x-axis at each x-intercept.
(c) Determine the maximum number of turning points on the graph.
(d) Determine the end behavior; that is, find the power function that the graph of f resembles for large values of |x|.

Learn how to analyze the polynomial function f(x) = (x - 7)^3 (x + 9)^2. Find real zeros, multiplicity, x-intercept behavior, maximum turning points, and end behavior. Perfect for high school algebra students.

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