Question

graph the polynomial function f(x)=x^{2}(x + 3) using parts (a) through (e). (v) 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 - intercept. the zero(s) of f 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 lesser zero of the function is of multiplicity , so the graph of f ▼ the x - axis at x = . the greater zero of the function is of multiplicity , so the graph of f ▼ the x - axis at x = .

Explanation:

Step1: Identify the zeros and their multiplicities

The polynomial is \(f(x)=x^{2}(x + 3)\). Setting \(f(x)=0\), we get \(x^{2}=0\) which gives \(x = 0\) with multiplicity 2, and \(x+3=0\) which gives \(x=-3\) with multiplicity 1. The lesser zero is \(x=-3\) and the greater zero is \(x = 0\).

Step2: Determine graph - x - axis behavior

If the multiplicity of a zero \(a\) of a polynomial function \(y = f(x)\) is odd, the graph of the function crosses the \(x\) - axis at \(x=a\). If the multiplicity is even, the graph of the function touches the \(x\) - axis at \(x=a\). Since the multiplicity of \(x=-3\) is 1 (odd), the graph of \(f\) crosses the \(x\) - axis at \(x=-3\). Since the multiplicity of \(x = 0\) is 2 (even), the graph of \(f\) touches the \(x\) - axis at \(x = 0\).

Answer:

The lesser zero of the function (\(x=-3\)) is of multiplicity 1, so the graph of \(f\) crosses the \(x\) - axis at \(x=-3\). The greater zero of the function (\(x = 0\)) is of multiplicity 2, so the graph of \(f\) touches the \(x\) - axis at \(x = 0\).