A polynomial function $p$ is given by $p(x)=-x(x-4)(x+2)$ . What are all intervals on which $p(x) \geq 0$ ? $\newline$ (A) $[-2,4]$ $\newline$ (B) $[-2,0] \cup[4, \infty)$ $\newline$ (C) $(-\infty,-4] \cup[0,2]$ $\newline$ (D) $(-\infty,-2] \cup[0,4]$

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Identify equivalent linear expressions: word problems

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Q. A polynomial function

p

is given by

p(x)=-x(x-4)(x+2)

. What are all intervals on which

p(x) \geq 0

\newline

(A)

[-2,4]

\newline

(B)

[-2,0] \cup[4, \infty)

\newline

(C)

(-\infty,-4] \cup[0,2]

\newline

(D)

(-\infty,-2] \cup[0,4]

Identify Zeros: Identify the zeros of the polynomial function $p(x)$ . $p(x) = -x(x - 4)(x + 2)$ has zeros at $x = 0$ , $x = 4$ , and $x = -2$ .
Determine Sign Intervals: Determine the sign of $p(x)$ on the intervals determined by the zeros. $\newline$ The zeros divide the number line into four intervals: $(-\infty, -2)$ , $(-2, 0)$ , $(0, 4)$ , and $(4, \infty)$ . We will test a point in each interval to determine if $p(x)$ is positive or negative in that interval.
Test Interval $-\infty$ to $-2$ : Test the interval $(-\infty, -2)$ by choosing a point less than $-2$ , for example, $x = -3$ .
$p(-3) = -(-3)((-3) - 4)((-3) + 2) = -3 \times (-7) \times (-1) = -21$ , which is negative.
So, $p(x)$ is negative on the interval $(-\infty, -2)$ .
Test Interval $-2$ to $0$ : Test the interval $(-2, 0)$ by choosing a point between $-2$ and $0$ , for example, $x = -1$ . $p(-1) = -(-1)((-1) - 4)((-1) + 2) = 1 \times (-5) \times 1 = -5$ , which is negative. $\newline$ So, $p(x)$ is negative on the interval $(-2, 0)$ .
Test Interval $0$ to $4$ : Test the interval $(0, 4)$ by choosing a point between $0$ and $4$ , for example, $x = 2$ . $p(2) = -(2)((2) - 4)((2) + 2) = -2 \times (-2) \times 4 = 16$ , which is positive. $\newline$ So, $p(x)$ is positive on the interval $(0, 4)$ .
Test Interval $4$ to $\infty$ : Test the interval $(4, \infty)$ by choosing a point greater than $4$ , for example, $x = 5$ . $p(5) = -(5)((5) - 4)((5) + 2) = -5 \times 1 \times 7 = -35$ , which is negative. $\newline$ So, $p(x)$ is negative on the interval $(4, \infty)$ .
Combine Positive Intervals: Combine the intervals where $p(x)$ is positive or zero. $\newline$ Since $p(x)$ is positive on the interval $(0, 4)$ , and it is zero at $x = 0$ and $x = 4$ , the interval on which $p(x)$ is greater than or equal to $0$ is $[0, 4]$ .