Question

graph the polynomial function f(x)=(x + 1)(x - 3)(x + 4) using parts (a) through (e). crosses or touches the x - axis at each x - intercept. the zero(s) of f is/are - 1,3, - 4. (simplify your answer. type an integer or a fraction. use a comma to separate answers as needed. type each answer only once.) the least zero is a zero of multiplicity , so the graph of f the x - axis at x = . the middle zero is a zero of multiplicity , so the graph of f the x - axis at x = . the greatest zero is a zero of multiplicity , so the graph of f the x - axis at x = .

Explanation:

Step1: Identify the zeros and their multiplicities

The polynomial is \(f(x)=(x + 1)(x - 3)(x + 4)\). Each factor \((x - a)\) gives a zero \(x=a\). Here, the zeros are \(x=-1,x = 3,x=-4\). Since each factor has an exponent of 1 (implicitly), the multiplicity of each zero is 1.

Step2: Determine the behavior at the x - intercepts

If the multiplicity of a zero \(a\) of a polynomial \(y = f(x)\) is odd, the graph of the function crosses the \(x\)-axis at \(x = a\). Since the multiplicity of \(x=-4\) (the least zero) is 1 (odd), the graph of \(f\) crosses the \(x\)-axis at \(x=-4\). The middle - zero \(x=-1\) has multiplicity 1 (odd), so the graph of \(f\) crosses the \(x\)-axis at \(x=-1\). The greatest zero \(x = 3\) has multiplicity 1 (odd), so the graph of \(f\) crosses the \(x\)-axis at \(x = 3\).

Answer:

The least zero is a zero of multiplicity 1, so the graph of \(f\) crosses the \(x\)-axis at \(x=-4\). The middle zero is a zero of multiplicity 1, so the graph of \(f\) crosses the \(x\)-axis at \(x=-1\). The greatest zero is a zero of multiplicity 1, so the graph of \(f\) crosses the \(x\)-axis at \(x = 3\).