Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. -(x - 3)(x - 4)^2 (x - 5) > 0

Verified step by step guidance

1. Step 1: First, we need to find the critical points of the inequality. The critical points are the values of x that make the inequality equal to zero. In this case, the critical points are x = 3, x = 4, and x = 5.
2. Step 2: Next, we need to test the intervals between the critical points. We can choose any number within each interval and substitute it into the inequality. If the inequality is true, then the entire interval is part of the solution set. If the inequality is false, then the interval is not part of the solution set.
3. Step 3: The intervals to test are (-∞, 3), (3, 4), (4, 5), and (5, ∞). Choose a test point in each interval and substitute it into the inequality. For example, you could choose x = 2 for the first interval, x = 3.5 for the second interval, x = 4.5 for the third interval, and x = 6 for the last interval.
4. Step 4: If the inequality is true for the test point, then the interval is part of the solution set. If the inequality is false for the test point, then the interval is not part of the solution set. Remember that because the inequality is '>', the critical points are not included in the solution set.
5. Step 5: Finally, write the solution set in interval notation. The solution set is the union of all the intervals that are part of the solution set.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to a value, typically zero, using inequality symbols such as >, <, ≥, or ≤. To solve these inequalities, one must determine the intervals where the polynomial is positive or negative, which often requires finding the roots of the polynomial and testing intervals between these roots.

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 (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2.

Sign Analysis

Sign analysis is a method used to determine the sign (positive or negative) of a polynomial expression over different intervals. After identifying the roots of the polynomial, one tests points in each interval to see if the polynomial evaluates to a positive or negative value, which helps in determining where the inequality holds true.