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Let's analyze the polynomial function \( f(x) = (x - 7)^3 (x + 9)^2 \) step by step to answer the questions.

### Given Information:
1. **Polynomial Function**: \( f(x) = (x - 7)^3 (x + 9)^2 \)
   
   This polynomial has two factors: \( (x - 7)^3 \) and \( (x + 9)^2 \).

### Part (a): List each real zero and its multiplicity.
   - The zeros of the function are values of \( x \) that make each factor zero.
   - **Zero at \( x = 7 \)**: From the factor \( (x - 7)^3 \), with multiplicity 3.
   - **Zero at \( x = -9 \)**: From the factor \( (x + 9)^2 \), with multiplicity 2.
   
   **Answer**: The smallest zero is \(-9\) with multiplicity \(2\), and the largest zero is \(7\) with multiplicity \(3\).

### Part (b): Determine whether the graph crosses or touches the x-axis at each x-intercept.
   - For **x = -9** (multiplicity 2), since it is an even multiplicity, the graph will **touch** the x-axis at this point.
   - For **x = 7** (multiplicity 3), since it is an odd multiplicity, the graph will **cross** the x-axis at this point.
   
   **Answer**: The graph **touches** the x-axis at \( x = -9 \) and **crosses** the x-axis at \( x = 7 \).

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

### Part (d): Determine the end behavior of \( f(x) \).
   - The highest degree term in \( f(x) = (x - 7)^3 (x + 9)^2 \) is \( x^5 \) (from expanding \( (x - 7)^3 (x + 9)^2 \) and taking the highest power).
   - The term \( x^5 \) determines the end behavior, which is similar to \( y = x^5 \).
   - Since \( x^5 \) has an odd degree and a positive leading coefficient, as \( x \to +\infty \), \( f(x) \to +\infty \); and as \( x \to -\infty \), \( f(x) \to -\infty \).
   
   **Answer**: The graph of \( f(x) \) resembles the power function \( y = x^5 \) for large values of \( |x| \).

If you need further explanation on any part or have specific questions, feel free to ask!

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### Related Practice Questions

1. How do you determine the multiplicity of each zero in a polynomial?
2. What is the difference in graph behavior between even and odd multiplicities at x-intercepts?
3. How does the degree of a polynomial relate to its end behavior?
4. How would the graph change if the polynomial had a factor with multiplicity 4 instead?
5. How do turning points relate to the maximum number of x-intercepts in a polynomial?

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### Tip
For end behavior, focus on the highest degree term in the polynomial, as it dictates the behavior as \( x \to \infty \) or \( x \to -\infty \).

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|.

Explore the polynomial function f(x) = (x - 7)^3 (x + 9)^2. Learn how to find its real zeros, determine graph behavior at each x-intercept, calculate maximum turning points, and analyze end behavior. This comprehensive breakdown is ideal for high school students studying polynomial functions.

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