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?

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.