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

Verified step by step guidance

1. First, write down the inequality clearly: \(x^3 + 2x^2 - x - 2 \geq 0\).
2. Next, factor the cubic polynomial on the left side. Start by grouping terms: \((x^3 + 2x^2) - (x + 2)\).
3. Factor each group: \(x^2(x + 2) - 1(x + 2)\), then factor out the common binomial: \((x + 2)(x^2 - 1)\).
4. Recognize that \(x^2 - 1\) is a difference of squares and factor it further: \((x + 2)(x - 1)(x + 1)\).
5. Set each factor equal to zero to find critical points: \(x + 2 = 0\), \(x - 1 = 0\), and \(x + 1 = 0\), which gives \(x = -2\), \(x = 1\), and \(x = -1\). Use these points to divide the number line into intervals and test the sign of the polynomial in each interval to determine where the inequality holds.

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) whose product equals the original polynomial. This step is crucial for solving polynomial inequalities because it helps identify the roots, which divide the number line into intervals to test for the inequality.

Interval Notation and Sign Analysis

Interval notation is a way to express solution sets using intervals on the real number line. After finding roots, sign analysis involves testing values in each interval to determine where the polynomial satisfies the inequality. The solution is then expressed using interval notation to clearly show all valid values.