To find the factors of the function [tex]\( f(x) = x^4 - 5x^3 - 4x^2 + 20x \)[/tex], we first express [tex]\( f(x) \)[/tex] in its factored form.
By factorizing, we get:
[tex]\[ f(x) = x(x - 5)(x - 2)(x + 2) \][/tex]
Next, we identify the roots (x-intercepts) of the function. These are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. From the factored form, it is clear that the roots are:
[tex]\[ x = -2, 0, 2, 5 \][/tex]
The graph of the function crosses the [tex]\( x \)[/tex]-axis at the points corresponding to these roots:
[tex]\[ (-2, 0), (0, 0), (2, 0), (5, 0) \][/tex]
Now, let's review the given statements and identify which one is true:
A. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (4, 0) \)[/tex].
- Incorrect. None of the roots are [tex]\( x = 4 \)[/tex].
B. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (-5, 0) \)[/tex].
- Incorrect. None of the roots are [tex]\( x = -5 \)[/tex].
C. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (2, 0) \)[/tex].
- Correct. One of the roots is [tex]\( x = 2 \)[/tex].
D. The graph crosses the [tex]\( x \)[/tex]-axis at the point [tex]\( (-4, 0) \)[/tex].
- Incorrect. None of the roots are [tex]\( x = -4 \)[/tex].
The correct answer is:
[tex]\[ \boxed{\text{C}} \][/tex]
