To solve this problem, we need to find the factors of the polynomial function [tex]\( f(x) = x^4 - 5x^3 - 4x^2 + 20x \)[/tex], and then use the factors to determine the points where the graph of the function crosses the [tex]\( x \)[/tex]-axis.
The factors of the given function [tex]\( f(x) \)[/tex] are:
[tex]\[ f(x) = x(x - 5)(x - 2)(x + 2) \][/tex]
From these factors, we can identify the roots of the function, which are the values of [tex]\( x \)[/tex] that make [tex]\( f(x) = 0 \)[/tex]. These roots are:
[tex]\[ x = 0, \][/tex]
[tex]\[ x = 5, \][/tex]
[tex]\[ x = 2, \][/tex]
[tex]\[ x = -2 \][/tex]
Based on these roots, the points where the graph of [tex]\( f(x) \)[/tex] crosses the [tex]\( x \)[/tex]-axis are:
[tex]\[ (0, 0), \][/tex]
[tex]\[ (5, 0), \][/tex]
[tex]\[ (2, 0), \][/tex]
[tex]\[ (-2, 0) \][/tex]
Now, let's analyze the given statements to determine which one is true:
A. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (2,0) \)[/tex].
- This statement is true because [tex]\( x = 2 \)[/tex] is one of the roots.
B. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (4,0) \)[/tex].
- This statement is false because [tex]\( x = 4 \)[/tex] is not one of the roots.
C. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (-5,0) \)[/tex].
- This statement is false because [tex]\( x = -5 \)[/tex] is not one of the roots.
D. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (-4,0) \)[/tex].
- This statement is false because [tex]\( x = -4 \)[/tex] is not one of the roots.
Therefore, the correct answer is:
A. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (2,0) \)[/tex].
