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. $x^3+2x^2-4x-8\ge 0$

Verified step by step guidance

1. First, write down the inequality clearly: \(x^3 + 2x^2 - 4x - 8 \geq 0\).
2. Next, factor the cubic polynomial on the left side. Start by looking for rational roots using the Rational Root Theorem, which suggests possible roots among the factors of the constant term (±1, ±2, ±4, ±8).
3. Once a root is found, use polynomial division or synthetic division to factor the cubic polynomial into a product of a linear factor and a quadratic factor.
4. After factoring completely, set each factor greater than or equal to zero to find critical points and determine intervals to test for the inequality.
5. Test values from each interval in the original inequality to determine where the expression is greater than or equal to zero, then express the solution set in interval notation and graph it on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols (>, <, ≥, ≤). 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.

Factoring Polynomials

Factoring is the process of breaking down a polynomial into simpler polynomials (factors) that multiply to give the original. It helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign in inequalities.

Interval Notation and Number Line Graphing

Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial satisfies the inequality, using open or closed dots to indicate whether endpoints are included.