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. Step 1: First, we need to factor the polynomial. The given polynomial is $x^4 - x^3 - 10x^2 - 8x$. We can factor out an $x$ to get $x(x^3 - x^2 - 10x - 8)$.
2. Step 2: Now, we need to find the roots of the polynomial. To do this, we set the polynomial equal to zero and solve for $x$. So, we have $x(x^3 - x^2 - 10x - 8) = 0$.
3. Step 3: The solutions to this equation are the values of $x$ that make the polynomial zero. These are the critical points of the inequality. We can find these by setting each factor equal to zero and solving for $x$.
4. Step 4: Once we have the critical points, we can use them to divide the number line into intervals. We then pick a test point from each interval and substitute it into the original inequality. If the inequality is true, then the entire interval is part of the solution set.
5. Step 5: Finally, we express the solution set in interval notation. This involves writing each interval as a pair of numbers in parentheses or brackets, with a comma between them. Parentheses are used for 'less than' and 'greater than' inequalities, while brackets are used for 'less than or equal to' and 'greater than or equal to' inequalities.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to a value, typically zero, using inequality symbols such as <, >, ≤, or ≥. To solve these inequalities, one must determine the intervals where the polynomial is either positive or negative. This often requires finding the roots of the polynomial and testing intervals between these roots.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2. This notation is essential for expressing solution sets of inequalities.

Sign Analysis

Sign analysis is a method used to determine the sign (positive or negative) of a polynomial across different intervals. After finding the roots of the polynomial, one tests points in each interval to see if the polynomial evaluates to a positive or negative value. This helps in identifying the intervals where the polynomial satisfies the inequality, which is crucial for providing the correct solution set.