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!

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?

Tip

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