Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. (x + 3)³(2x - 1)(x + 4) ≥ 0

Verified step by step guidance

1. Identify the critical points by setting each factor equal to zero: solve \( (x + 3)^3 = 0 \), \( 2x - 1 = 0 \), and \( x + 4 = 0 \).
2. Find the roots from each factor: \( x = -3 \) from \( (x + 3)^3 \), \( x = \frac{1}{2} \) from \( 2x - 1 \), and \( x = -4 \) from \( x + 4 \).
3. Plot these critical points on a number line to divide the real number line into intervals: \( (-\infty, -4) \), \( (-4, -3) \), \( (-3, \frac{1}{2}) \), and \( (\frac{1}{2}, \infty) \).
4. Choose a test point from each interval and substitute it into the inequality \( (x + 3)^3(2x - 1)(x + 4) \geq 0 \) to determine the sign of the expression in that interval.
5. Based on the sign test and the fact that the inequality includes \( \geq 0 \), include intervals where the expression is positive or zero, and write the solution set in interval notation.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality signs (>, <, ≥, ≤). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Critical Points and Sign Analysis

Critical points are the values of the variable that make any factor of the polynomial zero. These points divide the number line into intervals. By testing values from each interval, you determine whether the polynomial is positive or negative there, which helps identify the solution set.

Interval Notation

Interval notation is a concise way to represent sets of real numbers. It uses parentheses () for values not included and brackets [] for values included. After solving the inequality, the solution set is expressed in interval notation to clearly show all values satisfying the inequality.