Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. x⁴ + 6x² + 1 ≥ 4x³ + 4x

Verified step by step guidance

1. Rewrite the inequality by bringing all terms to one side to set the inequality to zero: \(x^{4} + 6x^{2} + 1 - 4x^{3} - 4x \geq 0\).
2. Rearrange the terms in descending order of powers of \(x\): \(x^{4} - 4x^{3} + 6x^{2} - 4x + 1 \geq 0\).
3. Attempt to factor the polynomial on the left side. Notice the pattern resembles a binomial expansion, so try to factor it as \((x^{2} - 2x + 1)^{2}\) or \((x - 1)^{4}\).
4. Confirm the factorization by expanding \((x - 1)^{4}\) to verify it matches the polynomial \(x^{4} - 4x^{3} + 6x^{2} - 4x + 1\).
5. Since \((x - 1)^{4} \geq 0\) for all real \(x\), determine the solution set by considering where the expression is greater than or equal to zero, which will be all real numbers.

Key Concepts

Polynomial Inequalities

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

Rearranging and Simplifying Expressions

To solve polynomial inequalities, first bring all terms to one side to set the inequality to zero. This simplification helps in factoring or applying other methods to determine where the polynomial is positive, negative, or zero, which is essential for identifying solution intervals.

Interval Notation and Sign Analysis

After finding critical points (roots), use sign analysis to test intervals between these points to see where the inequality holds. Expressing the solution set in interval notation concisely shows all values satisfying the inequality, using brackets for inclusive and parentheses for exclusive endpoints.