Analyze the polynomial function \( f(x)=x^{2}(x-3) \) using parts (a) through (e). (a) Determine the end behavior of the graph of the function. The graph of \( f \) behaves like \( y=x^{3} \) for large values of \( |x| \). (b) Find the \( x \) - and \( y \)-intercepts of the graph of the function. The \( x \)-intercept(s) is/are 0,3 . (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.) The \( y \)-intercept is 0 . (Simplify your answer. Type an integer or a fraction.) (c) Determine the zeros of the function and their multiplicity. Use this information to determine whether the graph crosses or touches the \( x \)-axis at each \( x \)-interce The zero(s) of \( f \) is/are \( \square \). (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed. Type each answer only once.)

We start with the given function:   f(x) = x²(x – 3) Part (a): End behavior The degree of f(x) is 3 (a cubic) and the leading term (after expansion) is x³. Therefore, for large |x|, the graph of f behaves like y = x³:  • As x → ∞, f(x) → ∞.  • As x → –∞, f(x) → –∞. Part (b): Intercepts • x-intercepts: Set f(x) = 0:   x²(x – 3) = 0 ⇒ x = 0 or x – 3 = 0 ⇒ x = 3.  Thus, the x-intercepts are x = 0 and x = 3. • y-intercept: Evaluate f(0):   f(0) = 0²(0 – 3) = 0.  Thus, the y-intercept is y = 0. Part (c): Zeros and their Multiplicities The function factors as f(x) = x²(x – 3), so the zeros are:  • x = 0 (this factor appears as x², so its multiplicity is 2).  • x = 3 (this factor appears to the power 1, so its multiplicity is 1). Interpretation regarding the x-axis:  • At x = 0, because the multiplicity is 2 (an even number), the graph touches the x-axis at this zero and bounces off.  • At x = 3, because the multiplicity is 1 (an odd number), the graph crosses the x-axis at this zero. Thus, the zeros of f(x) are 0 and 3; the graph touches at x = 0 and crosses at x = 3.