Question

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

Explanation:

Step1: Factor the function

$f(x)=x^3 - 4x=x(x^2 - 4)=x(x - 2)(x + 2)$

Step2: Find the zeros

Set $f(x)=0$, then $x(x - 2)(x + 2)=0$. By the zero - product property, if $ab = 0$, then $a = 0$ or $b = 0$. So $x=0$ or $x - 2=0$ or $x+2=0$. Solving these equations gives $x = 0,x=2,x=-2$.

Step3: Determine multiplicity

Each factor $(x),(x - 2),(x + 2)$ has an exponent of 1. So the multiplicity of $x=-2,0,2$ is 1. Since the multiplicity is odd, the graph crosses the $x$-axis at $x=-2,x = 0,x=2$.

Answer:

$-2,0,2$