Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. x⁵ + x² + 2 ≥ x⁴ + x³ + 2x

Verified step by step guidance

1. First, rewrite the inequality by bringing all terms to one side to set the inequality to zero: \(x^5 + x^2 + 2 - (x^4 + x^3 + 2x) \geq 0\).
2. Simplify the expression by combining like terms: \(x^5 - x^4 - x^3 + x^2 - 2x + 2 \geq 0\).
3. Next, factor the polynomial expression if possible. Start by looking for common factors or try polynomial division or synthetic division to find roots or factors.
4. Identify the critical points by solving the equation \(x^5 - x^4 - x^3 + x^2 - 2x + 2 = 0\). These points divide the number line into intervals to test.
5. Test values from each interval in the original inequality to determine where the expression is greater than or equal to zero, then write the solution set in interval notation based on these results.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another using inequality symbols (>, <, ≥, ≤). Solving them requires finding the values of the variable that make the inequality true, often by rearranging terms and analyzing the sign of the resulting polynomial.

Factoring and Simplifying Polynomials

To solve polynomial inequalities, it is essential to rewrite the inequality so that one side is zero. This often involves subtracting all terms to one side and factoring the resulting polynomial, which helps identify critical points where the polynomial changes sign.

Sign Analysis and Interval Notation

After factoring, sign analysis determines where the polynomial is positive or negative by testing values in intervals defined by the roots. The solution set is then expressed in interval notation, which concisely describes all values satisfying the inequality.