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Chapter 2: Problem 36

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$x^{3}+2 x^{2}-4 x-8 \geq 0$$

Short Answer

Expert verified
The solution set for the given polynomial inequality, in interval notation, is \((- \infty, -2] \cup [1, \infty)\)

Step by step solution

01

Factor the Polynomial

The given polynomial \(x^{3} +2x^{2} -4x -8\) factors as \((x - 1)(x + 2)(x + 2)\) . Therefore, the inequality to be solved becomes \((x - 1)(x + 2)^{2} \geq 0\)
02

Identify the 'Zeros'

Set each factor equal to zero and solve to find the zeros of the polynomial. The zeros are the solutions of the related polynomial equation. They are x = 1 and x = -2
03

Test Values in each Interval

The zeros divide the real number line into intervals. Choose a test value from each interval and substitute it into the inequality to determine where the inequality holds true. The intervals under consideration are \((- \infty, -2)\), \((-2,1)\) and \((1, \infty)\). For \((- \infty, -2)\), choose x = -3; the result of the inequality is positive, so this interval is part of the solution. For \((-2,1)\), choose x = 0; the result of the inequality is negative, so this interval is not part of the solution. For \((1, \infty)\), choose x = 2; the result of the inequality is positive, so this interval is part of the solution.

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