Solve each polynomial inequality. Give the solution set in interval notation. See Examples 2 and 3. x^4 + 2x^3 + 36 < 11x^2 + 12x

Verified step by step guidance

1. Step 1: First, we need to rearrange the inequality so that it equals zero. We do this by subtracting '11x^2 + 12x' from both sides of the inequality. This gives us 'x^4 + 2x^3 - 11x^2 - 12x + 36 < 0'.
2. Step 2: Next, we need to find the roots of the polynomial. This can be done by factoring the polynomial or using the Rational Root Theorem. If the polynomial is difficult to factor, you may need to use a graphing calculator to find the roots.
3. Step 3: Once we have the roots, 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 polynomial to check the sign of the result.
4. Step 4: If the result is negative, then the interval is part of the solution set. If the result is positive, then the interval is not part of the solution set.
5. Step 5: Finally, we write the solution set in interval notation. Remember that we use parentheses for 'less than' and 'greater than' inequalities, and brackets 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 using inequality symbols (e.g., <, >, ≤, ≥). To solve these inequalities, one typically rearranges the expression to set it to zero, allowing for the identification of critical points where the polynomial equals the value. The solution set is then determined by testing intervals around these critical points.

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, (a, b) represents all numbers between a and b, excluding a and b, while [a, b] includes both endpoints.

Critical Points

Critical points are values of the variable where the polynomial equals zero or is undefined. These points are essential in solving polynomial inequalities as they divide the number line into intervals. By testing the sign of the polynomial in each interval, one can determine where the inequality holds true, leading to the final solution set.