Answer :
Let's discuss the graph of the function [tex]\( g(x) = (x-1) (x+4)^3 (x+5)^2 \)[/tex] and determine at which points it crosses or touches the x-axis.
### Step-by-Step Solution:
1. First Term: [tex]\( (x-1) \)[/tex]
- The factor [tex]\( (x-1) \)[/tex] can be rewritten as [tex]\( x-1 \)[/tex].
- This factor is raised to the power of 1, which is an odd number.
- When a factor [tex]\( x-1 \)[/tex] is raised to an odd power, the graph crosses the x-axis at [tex]\( x = 1 \)[/tex].
2. Second Term: [tex]\( (x+4)^3 \)[/tex]
- The factor [tex]\( (x+4)^3 \)[/tex] can be rewritten as [tex]\( (x + 4) \)[/tex].
- This factor is raised to the power of 3, which is an odd number.
- When a factor [tex]\( x+4 \)[/tex] is raised to an odd power, the graph crosses the x-axis at [tex]\( x = -4 \)[/tex].
3. Third Term: [tex]\( (x+5)^2 \)[/tex]
- The factor [tex]\( (x+5)^2 \)[/tex] can be rewritten as [tex]\( (x + 5) \)[/tex].
- This factor is raised to the power of 2, which is an even number.
- When a factor [tex]\( x+5 \)[/tex] is raised to an even power, the graph only touches the x-axis at [tex]\( x = -5 \)[/tex].
Based on this analysis, we can conclude the following about the graph of [tex]\( g(x) \)[/tex]:
- The graph crosses the x-axis at [tex]\( (1, 0) \)[/tex].
- The graph crosses the x-axis at [tex]\( (-4, 0) \)[/tex].
- The graph touches the x-axis at [tex]\( (-5, 0) \)[/tex].
Thus, the completed statements describing the graph are:
- The graph crosses the axis at [tex]\( (1, 0) \)[/tex].
- The graph crosses the axis at [tex]\( (-4, 0) \)[/tex].
- The graph touches the axis at [tex]\( (-5, 0) \)[/tex].
### Step-by-Step Solution:
1. First Term: [tex]\( (x-1) \)[/tex]
- The factor [tex]\( (x-1) \)[/tex] can be rewritten as [tex]\( x-1 \)[/tex].
- This factor is raised to the power of 1, which is an odd number.
- When a factor [tex]\( x-1 \)[/tex] is raised to an odd power, the graph crosses the x-axis at [tex]\( x = 1 \)[/tex].
2. Second Term: [tex]\( (x+4)^3 \)[/tex]
- The factor [tex]\( (x+4)^3 \)[/tex] can be rewritten as [tex]\( (x + 4) \)[/tex].
- This factor is raised to the power of 3, which is an odd number.
- When a factor [tex]\( x+4 \)[/tex] is raised to an odd power, the graph crosses the x-axis at [tex]\( x = -4 \)[/tex].
3. Third Term: [tex]\( (x+5)^2 \)[/tex]
- The factor [tex]\( (x+5)^2 \)[/tex] can be rewritten as [tex]\( (x + 5) \)[/tex].
- This factor is raised to the power of 2, which is an even number.
- When a factor [tex]\( x+5 \)[/tex] is raised to an even power, the graph only touches the x-axis at [tex]\( x = -5 \)[/tex].
Based on this analysis, we can conclude the following about the graph of [tex]\( g(x) \)[/tex]:
- The graph crosses the x-axis at [tex]\( (1, 0) \)[/tex].
- The graph crosses the x-axis at [tex]\( (-4, 0) \)[/tex].
- The graph touches the x-axis at [tex]\( (-5, 0) \)[/tex].
Thus, the completed statements describing the graph are:
- The graph crosses the axis at [tex]\( (1, 0) \)[/tex].
- The graph crosses the axis at [tex]\( (-4, 0) \)[/tex].
- The graph touches the axis at [tex]\( (-5, 0) \)[/tex].
