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.