f(x)=x^(2)(x+2)(x-2)(x-5) has zeros at x=-2,x=0,x=2, and x=5. What is the sign of f on the interval -2

Question

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

Bytelearn · Accepted Answer

Identify Zeros: f(x) = x^2 * (x + 2) * (x - 2) * (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.

$f(x)=x^{2}(x+2)(x-2)(x-5)$ has zeros at $x=-2, x=0, x=2$ , and $x=5$ . $\newline$ What is the sign of $f$ on the interval \( -2

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f(x)=x2(x+2)(x−2)(x−5) f(x)=x^{2}(x+2)(x-2)(x-5) f(x)=x2(x+2)(x−2)(x−5) has zeros at x=−2,x=0,x=2 x=-2, x=0, x=2 x=−2,x=0,x=2, and x=5 x=5 x=5.\newlineWhat is the sign of f f f on the interval \( -2

$f(x)=x^{2}(x+2)(x-2)(x-5)$ has zeros at $x=-2, x=0, x=2$ , and $x=5$ . $\newline$ What is the sign of $f$ on the interval \( -2