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

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 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 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.