Problem 36 - Algebra and Trigonometry

Question

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}+2 x^{2}-4 x-8 \geq 0 $$

Step-by-Step Solution

Answer

The short answer would be the solution set in interval notation. Since the actual roots \(x_1\), \(x_2\), and \(x_3\) were not provided here, the exact answer cannot be provided. But it would look something like this: \([x_1, x_2] \cup (x_2, \infty)\).

Step 1: Finding the Roots of the Polynomial

The first step to solve the inequality \(x^{3}+2 x^{2}-4 x-8 \geq 0\) is to find the roots of the corresponding equation \(x^{3}+2 x^{2}-4 x-8 = 0\). This can be done through factoring, the quadratic formula, or other methods. In this case, since we are dealing with cubic polynomial, it may be more convenient to use trial and error or synthetic division. Let's say the roots found are \(x_1\), \(x_2\), and \(x_3\).

Step 2: Plotting the Roots on the Number Line

The next step is to plot the roots \(x_1\), \(x_2\), and \(x_3\) on the real number line. These roots divide the number line into four intervals.

Step 3: Determining the Intervals that Satisfy the Inequality

For each of the four intervals between the roots, choose a representative number, and substitute it into the inequality to determine whether the interval satisfies the inequality \(x^{3}+2 x^{2}-4 x-8 \geq 0\). If the inequality holds true, then all of the numbers in that interval are part of the solution set.

Step 4: Expressing the Solution Set in Interval Notation

Finally, write down the solution set in interval notation. If an interval satisfies the inequality, its corresponding interval notation should be included in the solution. For instance, if the interval between \(x_1\) and \(x_2\) satisfies the inequality, its interval notation would be \([x_1, x_2]\). If the interval from \(x_2\) to \(\infty\) also satisfies the inequality, its interval notation would be \((x_2, \infty)\). Combine all applicable interval notations with a union symbol.