We are given the function $f(x) = -2(x - 1)^2(x^2 - 25)$.

Step 1: Find the zeros of the function

To find the zeros of the function, we set $f(x) = 0$ and solve:

$-2(x - 1)^2(x^2 - 25) = 0$

This equation will be zero when either $(x - 1)^2 = 0$ or $x^2 - 25 = 0$.

For $(x - 1)^2 = 0$, we solve:

$x - 1 = 0 \quad \Rightarrow \quad x = 1$

For $x^2 - 25 = 0$, we solve:

$x^2 = 25 \quad \Rightarrow \quad x = 5 \text{ or } x = -5$

Step 2: Identify the multiplicities of the zeros

• The zero $x = 1$ comes from $(x - 1)^2$, so it has multiplicity 2.
• The zero $x = 5$ comes from $(x^2 - 25)$, so it has multiplicity 1.
• The zero $x = -5$ comes from $(x^2 - 25)$, so it has multiplicity 1.

Step 3: Determine how the graph interacts with the x-axis

• The graph touches the x-axis at $x = 1$ because the multiplicity of this zero is 2 (even multiplicity).
• The graph crosses the x-axis at $x = 5$ and $x = -5$ because the multiplicities of these zeros are 1 (odd multiplicity).

Step 4: Answer the questions

• The smallest real zero is $x = -5$, and it is a zero of multiplicity 1, so the graph crosses the x-axis at $x = -5$.
• The middle real zero is $x = 1$, and it is a zero of multiplicity 2, so the graph touches the x-axis at $x = 1$.
• The largest real zero is $x = 5$, and it is a zero of multiplicity 1, so the graph crosses the x-axis at $x = 5$.

Summary of Answers:

• The smallest real zero is $x = -5$, with a multiplicity of 1, so the graph crosses at $x = -5$.
• The middle real zero is $x = 1$, with a multiplicity of 2, so the graph touches at $x = 1$.
• The largest real zero is $x = 5$, with a multiplicity of 1, so the graph crosses at $x = 5$.

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Would you like more details on the behavior of the graph or any other concepts? Here are some related questions to expand on the topic:

1. How does the multiplicity of a zero affect the shape of the graph at that zero?
2. How do we factor polynomials to find the zeros and their multiplicities?
3. How does the sign of the leading coefficient affect the direction of the graph at extreme values of $x$?
4. What are other ways to solve for the zeros of a polynomial besides factoring?
5. How would the graph change if the coefficient $-2$ was replaced with a positive value?

Tip: When determining the behavior of a graph near a zero, check the multiplicity—if the multiplicity is odd, the graph crosses the x-axis; if even, it touches and bounces back!