$f(x)=x^{2}(x+2)(x-2)(x-5)$ has zeros at $x=-2, x=0, x=2$ , and $x=5$ . $\newline$ What is the sign of $f$ on the interval $-2<x<5$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $f$ is always positive on the interval. $\newline$ (B) $f$ is always negative on the interval. $\newline$ (C) $f$ is sometimes positive and sometimes negative on the interval.

Full solution

f(x)=x^{2}(x+2)(x-2)(x-5)

has zeros at

x=-2, x=0, x=2

, and

x=5

\newline

What is the sign of

f

on the interval

-2<x<5

\newline

Choose

1

answer:

\newline

(A)

f

is always positive on the interval.

\newline

(B)

f

is always negative on the interval.

\newline

(C)

f

is sometimes positive and sometimes negative on the interval.

Identify Zeros: $f(x) = x^2 \cdot (x + 2) \cdot (x - 2) \cdot (x - 5)$ has zeros at $x = -2$ , $x = 0$ , $x = 2$ , and $x = 5$ . To find the sign of $f$ on the interval $-2 < x < 5$ , we need to look at the sign of each factor in the interval.
Analyze $x^2$ : For $x^2$ , any value of $x$ (except $0$ ) will give a positive result since squaring a number always results in a positive number.
Analyze $(x + 2)$ : For $(x + 2)$ , any value of $x$ greater than $-2$ will make this factor positive.
Analyze $(x - 2)$ : For $(x - 2)$ , any value of $x$ greater than $2$ will make this factor positive, but since we're looking at the interval $-2 < x < 5$ , this factor will be negative for $-2 < x < 2$ .
Analyze $(x - 5)$ : For $(x - 5)$ , any value of $x$ less than $5$ will make this factor negative, which applies to the entire interval $-2 < x < 5$ .
Calculate Sign $-2 < x < 2$ : Now, let's multiply the signs of each factor together for the interval $-2 < x < 2$ . We have a positive from $x^2$ , a positive from $(x + 2)$ , a negative from $(x - 2)$ , and a negative from $(x - 5)$ . Positive times positive times negative times negative equals positive.
Calculate Sign $2 < x < 5$ : Next, for the interval $2 < x < 5$ , we have a positive from $x^2$ , a positive from $(x + 2)$ , a positive from $(x - 2)$ , and a negative from $(x - 5)$ . Positive times positive times positive times negative equals negative.