A polynomial function p is given by p(x)=-x(x-4)(x+2). What are all intervals on which p(x) = 0 ? (A) [-2,4] (B) [-2,0]uu[4,oo) (C) (-oo,-4]uu[0,2] (D) (-oo,-2]uu[0,4]

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A polynomial function p is given by p(x)=-x(x-4)(x+2). What are all intervals on which p(x) >= 0 ? (A) [-2,4] (B) [-2,0]uu[4,oo) (C) (-oo,-4]uu[0,2] (D) (-oo,-2]uu[0,4]

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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. The zeros divide the number line into four intervals: (-∞, -2), (-2, 0), (0, 4), and (4, ∞). We will test a point in each interval to determine if p(x) is positive or negative in that interval.Test Interval -∞ to -2: Test the interval (-∞, -2) by choosing a point less than -2, for example, x = -3. p(-3) = -(-3)((-3) - 4)((-3) + 2) = -3 * (-7) * (-1) = -21, which is negative. So, p(x) is negative on the interval (-∞, -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 * (-5) * 1 = -5, which is negative. 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 * (-2) * 4 = 16, which is positive. So, p(x) is positive on the interval (0, 4).Test Interval 4 to ∞: Test the interval (4, ∞) by choosing a point greater than 4, for example, x = 5. p(5) = -(5)((5) - 4)((5) + 2) = -5 * 1 * 7 = -35, which is negative. So, p(x) is negative on the interval (4, ∞).Combine Positive Intervals: Combine the intervals where p(x) is positive or zero. 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].

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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