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. 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. Combine like terms to simplify the expression: \(x^{4} + 2x^{3} - 11x^{2} - 12x + 36 < 0\).
3. Attempt to factor the polynomial \(x^{4} + 2x^{3} - 11x^{2} - 12x + 36\) by grouping or using substitution methods to find its roots.
4. Find the critical points by solving the equation \(x^{4} + 2x^{3} - 11x^{2} - 12x + 36 = 0\); these points divide the number line into intervals.
5. Test values from each interval in the original inequality to determine where the polynomial is less than zero, then express the solution set in interval notation.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another value or polynomial using inequality signs (<, >, ≤, ≥). Solving them requires finding the values of the variable that make the inequality true, often by rewriting the inequality in standard form and analyzing the sign of the polynomial.

Factoring and Simplifying Polynomials

To solve polynomial inequalities, it is essential to rewrite the inequality so one side equals zero. This often involves moving all terms to one side and factoring the polynomial if possible. Factoring helps identify critical points where the polynomial changes sign, which are key to determining solution intervals.

Sign Analysis and Interval Notation

After finding critical points from the factored polynomial, sign analysis determines where the polynomial is positive or negative by testing values in each interval. The solution set is then expressed in interval notation, which concisely describes all values of the variable that satisfy the inequality.