Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. 23. (a) -x(x - 1)(x - 2) ≥ 0 (b) -x(x - 1)(x - 2) > 0 (c) -x(x - 1)(x - 2) ≤ 0 (d) -x(x - 1)(x - 2) < 0

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero using inequality signs (≥, >, ≤, <). To solve these inequalities, one must determine the intervals where the polynomial is positive or negative. This requires finding the roots of the polynomial and testing intervals between these roots to see where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval [a, b) includes 'a' but not 'b', while (a, b) excludes both endpoints. This notation is essential for expressing solution sets of inequalities succinctly.

Test Points Method

The test points method is a strategy used to determine the sign of a polynomial in each interval created by its roots. After identifying the roots, one selects a test point from each interval and substitutes it into the polynomial. The sign of the result indicates whether the polynomial is positive or negative in that interval, which is crucial for solving inequalities.