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+2x^2−4x−8≥0

Verified step by step guidance

1. <Step 1: Identify the polynomial inequality. The given inequality is \(x^3 + 2x^2 - 4x - 8 \geq 0\).>
2. <Step 2: Factor the polynomial, if possible. Look for common factors or use techniques like synthetic division or the Rational Root Theorem to find roots.>
3. <Step 3: Determine the critical points by setting the polynomial equal to zero and solving for \(x\). These points will help divide the number line into intervals.>
4. <Step 4: Test each interval by selecting a test point from each interval and substituting it back into the original inequality to determine if the interval satisfies the inequality.>
5. <Step 5: Express the solution set in interval notation, including endpoints where the inequality is \(\geq\) or \(\leq\), and graph the solution set on a real number line.>

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to a value using inequality symbols (e.g., ≥, ≤, >, <). To solve these inequalities, one typically finds the roots of the polynomial, determines the intervals on the number line, and tests these intervals to see where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, [a, b] includes both a and b, while (a, b) does not.

Graphing Solution Sets

Graphing solution sets involves visually representing the solutions of an inequality on a number line. This includes marking the critical points (roots) and shading the regions that satisfy the inequality. Understanding how to accurately depict these regions is crucial for interpreting the solution set effectively.