Question

graph the polynomial function f(x)=(x - 1)(x + 4)^2 using parts (a) through (e). (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 2, so the graph of f touches the x - axis at x=-4. the greater zero of the function is of multiplicity 1, so the graph of f crosses the x - axis at x=1. (d) determine the maximum number of turning points on the graph of the function. (type a whole number.)

Explanation:

Step1: Recall polynomial - degree formula

For a polynomial \(f(x)=(x - a_1)^{n_1}(x - a_2)^{n_2}\cdots(x - a_k)^{n_k}\), the degree \(n=n_1 + n_2+\cdots + n_k\). Here, \(f(x)=(x - 1)(x + 4)^2\), so \(n=1 + 2=3\).

Step2: Recall turning - point formula

The maximum number of turning points of a polynomial function of degree \(n\) is \(n - 1\). Since \(n = 3\), the maximum number of turning points is \(3-1=2\).

Answer:

2