Final answer:
The polynomial function f(x) = x^4 - x^3 + x^2 - x has at least one x-intercept at x = 0, and the maximum number of x-intercepts is four. Without further analysis or graphing, we cannot determine the exact number of x-intercepts beyond the guaranteed one.
Step-by-step explanation:
The graph of the polynomial function f(x) = x⁴ - x³ + x² - x can have a maximum number of x-intercepts equal to the degree of the polynomial, which in this case is 4. However, to determine the exact number of x-intercepts, we can factor the polynomial. Factoring out an x, we get x(x³ - x² + x - 1). The remaining cubic factor does not have rational roots, and we need further analysis or graphing to determine the exact count. Nevertheless, there is at least one x-intercept since we can see that x = 0 is a solution to the equation. Thus, without additional information, we cannot determine the exact number of x-intercepts, but we can confirm there is at least one. The possible choices for the answer are:
- A) 1 x-intercept
- B) 2 x-intercepts
- C) 3 x-intercepts
- D) 4 x-intercepts
Without a concrete method to ascertain the specific number of x-intercepts beyond the guaranteed one at x = 0, we should attempt to either graph the function or apply further algebraic methods to reveal all the real roots.