Let’s analyze the given polynomial function step by step.

Given Function:

$f(x) = (x - 2)^2 (x + 9)^3$

(a) List each real zero and its multiplicity:

The zeros of a polynomial are the values of $x$ that make $f(x) = 0$.

1. From the factor $(x - 2)^2$, $x = 2$ is a zero with multiplicity $2$.
2. From the factor $(x + 9)^3$, $x = -9$ is a zero with multiplicity $3$.

Zeros and their multiplicities:

• $x = 2$, multiplicity $2$
• $x = -9$, multiplicity $3$

(b) Determine whether the graph crosses or touches the x-axis at each x-intercept:

The behavior at each intercept is determined by the multiplicity:

1. Multiplicity 2 (even): The graph touches the x-axis at $x = 2$ but does not cross it.
2. Multiplicity 3 (odd): The graph crosses the x-axis at $x = -9$.

Summary:

• At $x = 2$, the graph touches the x-axis.
• At $x = -9$, the graph crosses the x-axis.

(c) Determine the end behavior:

The end behavior of a polynomial is determined by the degree and the leading term.

1. Degree of the polynomial: $\text{Degree} = 2 (from$(x - 2)^2$) + 3 (from$(x + 9)^3$) = 5$

   The degree is odd.

2. Leading coefficient: The product of the leading terms in $(x - 2)^2$ and $(x + 9)^3$ is positive: $(+1)(+1)^5 = +1$

3. End behavior for odd-degree polynomials with a positive leading coefficient:

   • As $x \to -\infty$, $f(x) \to -\infty$ (falls to the left).
   • As $x \to +\infty$, $f(x) \to +\infty$ (rises to the right).

Answer for end behavior: B. Falls left and rises right.

Final Answer Summary:

(a) Real zeros:

• $x = 2$, multiplicity 2
• $x = -9$, multiplicity 3

(b) Behavior at intercepts:

• Touches the x-axis at $x = 2$.
• Crosses the x-axis at $x = -9$.

(c) End behavior:

• Falls left and rises right (B).

Do you have further questions or would you like a detailed graph analysis? Here are 5 related questions:

1. How can you confirm the behavior of a graph at a zero using calculus (e.g., derivatives)?
2. Why does even multiplicity result in touching but odd multiplicity results in crossing the x-axis?
3. How do you determine leading coefficients from factored forms?
4. What changes in end behavior if the leading coefficient is negative?
5. How would adding another factor (e.g., $(x + 5)^2$) affect the end behavior?

Tip: Always verify graph behaviors by analyzing the polynomial's degree, leading coefficient, and multiplicities!