To find the domain of a function, we need to identify any values of x that would result in undefined expressions. In this case, the function f(x) contains only algebraic operations, so the domain of f(x) is all real numbers, or (-∞, ∞).
To find the real zeros of the function, we need to solve for the values of x that make the expression equal to zero. So we start by setting f(x) = 0 and then simplifying the resulting equation:
f(x) = (x-3)^2 x³ - 3x² + 2x = 0
Factor out an x from the right-hand side of the equation:
x (x-3)^2 (x² - 3x + 2) = 0
The real zeros of the function are the values of x that make the left-hand side of the equation equal to zero. So we solve each factor for x:
x = 0, x = 3, and x² - 3x + 2 = 0
The quadratic equation x² - 3x + 2 = 0 can be factored as (x-1)(x-2) = 0, so the solutions are x = 1 and x = 2.
Therefore, the real zeros of the function are x = 0, x = 1, x = 2, and x = 3.