To understand the behavior of the graph of [tex]\( f(x) = 4x^7 + 40x^6 + 100x^5 \)[/tex] and how it interacts with the [tex]\( x \)[/tex]-axis, let's address the critical points where the graph meets or touches the [tex]\( x \)[/tex]-axis.
1. Finding the Roots:
The function [tex]\( f(x) = 4x^7 + 40x^6 + 100x^5 \)[/tex] has roots at [tex]\( x = -5 \)[/tex] and [tex]\( x = 0 \)[/tex].
2. Behavior at [tex]\( x = 0 \)[/tex]:
To determine whether [tex]\( f(x) \)[/tex] crosses or just touches the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex], we need to look at the multiplicity of the root.
The term [tex]\( x^5 \)[/tex] suggests that [tex]\( x = 0 \)[/tex] is a root with multiplicity 5. Generally, when a polynomial has a root with an odd multiplicity, the graph crosses the [tex]\( x \)[/tex]-axis at that root. Since 5 is odd, the graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex].
3. Behavior at [tex]\( x = -5 \)[/tex]:
Similarly, to determine the behavior at [tex]\( x = -5 \)[/tex], we look at the multiplicity. Here, [tex]\( x = -5 \)[/tex] comes from the term involving [tex]\( x^6 \)[/tex], which suggests potential multiplicity 1 for [tex]\( x = -5 \)[/tex]. Since 1 is also odd, the graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = -5 \)[/tex].
Given the above points, we conclude:
- The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex].
- The graph touches the [tex]\( x \)[/tex]-axis at [tex]\( x = -5 \)[/tex].
Therefore, the correct description of the graph is:
"The graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 0 \)[/tex] and touches the [tex]\( x \)[/tex]-axis at [tex]\( x = -5 \)[/tex]."
