Question

consider the following polynomial inequality.
$x^2(x + 4)(x - 2) > 0$
step 2 of 2: test each interval to find the solution of the polynomial inequality.
express your answer in interval notation.
answer 4 points
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Explanation:

Step1: Find critical points

Set each factor equal to zero: \(x^2 = 0\) gives \(x = 0\), \(x + 4 = 0\) gives \(x=-4\), \(x - 2 = 0\) gives \(x = 2\). Critical points are \(x=-4\), \(x = 0\), \(x = 2\). These divide the number line into intervals: \((-\infty, -4)\), \((-4, 0)\), \((0, 2)\), \((2, \infty)\).

Step2: Test intervals

• For \((-\infty, -4)\), pick \(x=-5\): \((-5)^2(-5 + 4)(-5 - 2)=25\times(-1)\times(-7)=175>0\), so this interval is part of the solution.
• For \((-4, 0)\), pick \(x=-1\): \((-1)^2(-1 + 4)(-1 - 2)=1\times3\times(-3)=-9<0\), not part of solution.
• For \((0, 2)\), pick \(x = 1\): \(1^2(1 + 4)(1 - 2)=1\times5\times(-1)=-5<0\), not part of solution.
• For \((2, \infty)\), pick \(x = 3\): \(3^2(3 + 4)(3 - 2)=9\times7\times1 = 63>0\), so this interval is part of the solution.

Note that \(x = 0\), \(x=-4\), \(x = 2\) make the left - hand side zero or undefined (but here it's zero), so they are not included.

Answer:

\((-\infty, - 4)\cup(2, \infty)\)