Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. x⁴ + 2x³ + 36 < 11x² + 12x

Verified step by step guidance

1. First, rewrite the inequality by bringing all terms to one side to set the inequality to zero: \(x^4 + 2x^3 + 36 - 11x^2 - 12x < 0\).
2. Simplify the expression by combining like terms: \(x^4 + 2x^3 - 11x^2 - 12x + 36 < 0\).
3. Next, attempt to factor the polynomial \(x^4 + 2x^3 - 11x^2 - 12x + 36\) by grouping or using other factoring techniques such as synthetic division or the Rational Root Theorem to find possible roots.
4. Once factored, identify the critical points (roots) where the expression equals zero. These points divide the number line into intervals.
5. Test a value from each interval in the original inequality to determine where the polynomial is less than zero, then express the solution set using interval notation based on these results.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another using inequality symbols like <, >, ≤, or ≥. Solving them requires finding the values of the variable that make the inequality true, often by rewriting the inequality so one side is zero and analyzing the sign of the resulting polynomial.

Factoring and Simplifying Polynomials

To solve polynomial inequalities, it is essential to simplify the inequality by moving all terms to one side and factoring the polynomial if possible. Factoring breaks the polynomial into products of simpler expressions, which helps identify critical points where the polynomial changes sign.

Sign Analysis and Interval Notation

After finding the critical points (roots), sign analysis determines where the polynomial is positive or negative by testing values in intervals between roots. The solution set is then expressed in interval notation, which concisely represents all values satisfying the inequality.