Final answer:
The zeros of f(x) = x^3 - 3x^2 + 2x are 0, 1, and 2 with multiplicities of 1 and 2. The graph crosses the x-axis at x = 0 and touches or bounces off the x-axis at x = 1 and x = 2.
Step-by-step explanation:
The zeros of the function f(x) = x^3 - 3x^2 + 2x can be found by setting the function equal to zero and solving for x. In this case, we have x^3 - 3x^2 + 2x = 0. Factoring out an x, we get x(x^2 - 3x + 2) = 0. Setting each factor equal to zero, we find that the zeros are x = 0, x = 1, and x = 2.
The multiplicities of the zeros can be determined by examining the degree of each factor. Since the factor x has a degree of 1, its multiplicity is 1. The factor (x^2 - 3x + 2) can be factored further as (x - 1)(x - 2), so its degree is 2. Therefore, its multiplicity is also 2.
The graph behaves differently at each zero depending on its multiplicity. At x = 0, the graph crosses the x-axis. At x = 1 and x = 2, the graph touches or bounces off the x-axis, but does not cross it. This is because the multiplicity of each zero determines the behavior of the graph: odd multiplicities result in crossing the x-axis, while even multiplicities result in touching or bouncing off the x-axis.