Solve each polynomial inequality. Give the solution set in interval notation. x⁴ - x³ - 10x² - 8x < 0

Verified step by step guidance

1. First, rewrite the inequality clearly: \(x^4 - x^3 - 10x^2 - 8x < 0\).
2. Factor the polynomial expression on the left side. Start by 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, write the inequality as a product of linear (or irreducible) factors set less than zero, for example: \(x (x - a)(x - b)(x - c) < 0\), where \(a\), \(b\), and \(c\) are the roots found.
5. Determine the sign of the product on intervals defined by the roots by testing points in each interval. Use this to identify where the product is negative, and express the solution set in interval notation accordingly.

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. It helps identify the roots or zeros of the polynomial, which are critical points where the sign of the polynomial can change, aiding in solving inequalities.

Sign Analysis and Interval Notation

Sign analysis involves testing intervals determined by the polynomial's roots to determine 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.