To find the factors of the function [tex]\( f(x) = x^4 - 5x^3 - 4x^2 + 20x \)[/tex], we need to perform factorization.
Firstly, let's write the function clearly:
[tex]\[ f(x) = x^4 - 5x^3 - 4x^2 + 20x \][/tex]
Through factorization, we can express [tex]\( f(x) \)[/tex] as a product of its factors:
[tex]\[ f(x) = x(x - 5)(x - 2)(x + 2) \][/tex]
Next, to understand where the graph of the function crosses the [tex]\( x \)[/tex]-axis, we set the function equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x(x - 5)(x - 2)(x + 2) = 0 \][/tex]
Setting each factor equal to zero gives us the roots of the function:
[tex]\[ x = 0 \][/tex]
[tex]\[ x - 5 = 0 \Rightarrow x = 5 \][/tex]
[tex]\[ x - 2 = 0 \Rightarrow x = 2 \][/tex]
[tex]\[ x + 2 = 0 \Rightarrow x = -2 \][/tex]
These roots indicate the points where the graph of the function [tex]\( f(x) \)[/tex] crosses the [tex]\( x \)[/tex]-axis. Thus, the graph crosses the [tex]\( x \)[/tex]-axis at the points:
[tex]\[ (0, 0), (5, 0), (2, 0), (-2, 0) \][/tex]
Considering the choices given:
A. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (4,0) \)[/tex].
B. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (-5,0) \)[/tex].
C. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (-4,0) \)[/tex].
D. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (2,0) \)[/tex].
From the list above, the correct statement is:
D. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (2,0) \)[/tex].
