Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. x^4 - x^3 - 10x^2 - 8x < 0

Verified step by step guidance

1. First, rewrite the inequality: \(x^4 - x^3 - 10x^2 - 8x < 0\).
2. Factor the polynomial on the left side by first factoring out the greatest common factor (GCF), which is \(x\): \(x(x^3 - x^2 - 10x - 8) < 0\).
3. Next, factor the cubic polynomial \(x^3 - x^2 - 10x - 8\). Use methods such as the Rational Root Theorem to find possible roots and then perform polynomial division or synthetic division to factor it completely.
4. Once fully factored, express the inequality as a product of linear and/or quadratic factors set less than zero: \((x)( ext{factor}_1)( ext{factor}_2)( ext{factor}_3) < 0\).
5. Determine the critical points by setting each factor equal to zero, then use a sign chart to test intervals between these points to find where the product is negative. Finally, write the solution set in interval notation based on these intervals.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality signs. 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 expressing a polynomial as a product of simpler polynomials or factors. It helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign in inequality problems.

Sign Analysis and Interval Notation

Sign analysis involves testing intervals determined by the polynomial's roots to see where the polynomial is positive or negative. Interval notation is a concise way to express the solution set, showing all values of the variable that satisfy the inequality.