Question

graph the polynomial function f(x)=x^3 - 4x. answer parts (a) through (e).
the zero(s) of f is/are -2,0,2.
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)
the smallest 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 largest zero is a zero of multiplicity , so the graph of f the x - axis at x = .

Explanation:

Step1: Factor the polynomial

We have $f(x)=x^{3}-4x = x(x^{2}-4)=x(x - 2)(x + 2)$. The zeros are found by setting $f(x)=0$, so $x(x - 2)(x + 2)=0$, which gives $x=-2,0,2$. The smallest zero is $x=-2$, the middle zero is $x = 0$, and the largest zero is $x=2$.
For a zero $r$ of a polynomial $y = f(x)$, if $(x - r)^n$ is a factor of $f(x)$ and $(x - r)^{n+1}$ is not a factor, then the multiplicity of the zero $r$ is $n$. Here, each factor $(x+2)$, $x$, and $(x - 2)$ has an exponent of 1.

Step2: Determine the behavior at the zeros

If the multiplicity of a zero $r$ is odd, the graph of the polynomial crosses the $x$-axis at $x = r$. Since the multiplicity of $x=-2$ is 1 (odd), the graph of $f$ crosses the $x$-axis at $x=-2$. Since the multiplicity of $x = 0$ is 1 (odd), the graph of $f$ crosses the $x$-axis at $x = 0$. Since the multiplicity of $x=2$ is 1 (odd), the graph of $f$ crosses the $x$-axis at $x=2$.

Answer:

The smallest zero is a zero of multiplicity 1, so the graph of $f$ crosses the $x$-axis at $x=-2$. The middle zero is a zero of multiplicity 1, so the graph of $f$ crosses the $x$-axis at $x = 0$. The largest zero is a zero of multiplicity 1, so the graph of $f$ crosses the $x$-axis at $x=2$.