Final answer:
The function f(x) = x(x-4)²(x - 5)³ has exactly three x-intercepts, corresponding to its three distinct roots x = 0, x = 4, and x = 5, despite some roots having multiplicities greater than one. Therefore, the answer is B) 3.
Step-by-step explanation:
The function f(x) = x(x-4)²(x - 5)³ has x-intercepts at the points where f(x) = 0. In this case, that occurs when x = 0, x = 4, and x = 5. Each root corresponds to an x-intercept; however, the multiplicity of the root indicates if the graph touches or crosses the x-axis at that intercept. Here, x = 0 is a simple root, x = 4 is a double root (touching at the intercept), and x = 5 is a triple root (also touching at the intercept).Consequently, there are exactly three x-intercepts for this function corresponding to its three distinct roots, even though two of these roots have multiplicity greater than one. Therefore, the correct option is B) 3.