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To solve this problem, let's analyze the function \( f(x) = 4x^7 + 40x^6 + 100x^5 \) and determine its behavior at the x-intercepts.

### Step 1: Factor the Polynomial
Let's first factor out the common term:
\[
f(x) = 4x^5(x^2 + 10x + 25)
\]
Now, simplify the quadratic part:
\[
x^2 + 10x + 25 = (x + 5)^2
\]
This gives:
\[
f(x) = 4x^5(x + 5)^2
\]

### Step 2: Find the x-intercepts
For \( f(x) = 0 \):
1. \( x^5 = 0 \) implies \( x = 0 \).
2. \( (x + 5)^2 = 0 \) implies \( x = -5 \).

Thus, the x-intercepts are at \( x = 0 \) and \( x = -5 \).

### Step 3: Analyze the Behavior at Each Intercept
1. **At \( x = 0 \):** The factor \( x^5 \) has an odd exponent (5), which means the graph **crosses** the x-axis at \( x = 0 \).
2. **At \( x = -5 \):** The factor \( (x + 5)^2 \) has an even exponent (2), which means the graph **touches** the x-axis at \( x = -5 \).

### Conclusion
The correct answer is:
- **The graph crosses the x-axis at \( x = 0 \) and touches the x-axis at \( x = -5 \).**

This corresponds to the third option:
- **"The graph crosses the x-axis at \( x = 0 \) and touches the x-axis at \( x = -5 \)."**

Would you like further details or explanations on any part of this solution?

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Here are five related questions to expand on this information:
1. What is the significance of an odd versus even exponent in determining whether a graph crosses or touches the x-axis?
2. How would the behavior change if the factor \( x^5 \) were \( x^6 \) instead?
3. What happens to the graph if we alter the polynomial to \( 4x^7 + 40x^6 + 100x^5 + c \) where \( c \neq 0 \)?
4. How can we determine if a function has multiple roots without fully factoring it?
5. What are the general steps to analyze the behavior of polynomial functions at their roots?

**Tip:** Remember that an odd power at a root causes the graph to cross the x-axis, while an even power causes it to just touch the axis.

Which statement describes the graph of f(x) = 4x^7 + 40x^6 + 100x^5?

Learn how to analyze the behavior of the graph for the polynomial function f(x) = 4x^7 + 40x^6 + 100x^5. This solution includes step-by-step factoring, identification of x-intercepts, and explanation of graph behavior at each root. Suitable for high school students in Grades 10-12.

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