To determine the number of [tex]\( x \)[/tex]-intercepts of the polynomial function [tex]\( f(x) = x^4 - x^3 + x^2 - x \)[/tex], we need to solve for [tex]\( x \)[/tex] when [tex]\( f(x) = 0 \)[/tex].
We start by factoring the polynomial:
[tex]\[
f(x) = x^4 - x^3 + x^2 - x = x(x^3 - x^2 + x - 1)
\][/tex]
Next, we need to solve the equation,
[tex]\[
x(x^3 - x^2 + x - 1) = 0
\][/tex]
This gives two cases:
1. [tex]\( x = 0 \)[/tex]
2. [tex]\( x^3 - x^2 + x - 1 = 0 \)[/tex]
For the second case, let's solve the cubic equation [tex]\( x^3 - x^2 + x - 1 = 0 \)[/tex].
The solutions to this cubic equation are:
[tex]\[
x = 1, \quad x = -i, \quad x = i
\][/tex]
where [tex]\( i \)[/tex] is the imaginary unit ([tex]\( i^2 = -1 \)[/tex]).
Thus, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[
0, \, 1, \, -i, \, i
\][/tex]
However, we are specifically interested in the real [tex]\( x \)[/tex]-intercepts for the graph, and only real numbers can be [tex]\( x \)[/tex]-intercepts on the Cartesian plane. From the above solutions, the real [tex]\( x \)[/tex]-intercepts are:
[tex]\[
0, \, 1
\][/tex]
This means there are two real [tex]\( x \)[/tex]-intercepts.
Therefore, the number of [tex]\( x \)[/tex]-intercepts on the graph of the polynomial function [tex]\( f(x) = x^4 - x^3 + x^2 - x \)[/tex] are:
[tex]\[
2 \, \text{real \( x \)-intercepts}
\][/tex]
So out of the given options, the correct answer is:
[tex]\[
2 \, \text{x-intercepts}
\][/tex]
