Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3+x^2+4x+4>0

Verified step by step guidance

1. Step 1: Start by factoring the polynomial \(x^3 + x^2 + 4x + 4\). Look for common factors or use techniques like grouping.
2. Step 2: Once factored, identify the critical points by setting each factor equal to zero and solving for \(x\). These points will help determine the intervals to test.
3. Step 3: Use the critical points to divide the real number line into intervals. For each interval, choose a test point and substitute it into the inequality to determine if the interval satisfies the inequality.
4. Step 4: Analyze the sign of the polynomial in each interval. If the test point makes the inequality true, that interval is part of the solution set.
5. Step 5: Express the solution set in interval notation, combining all intervals where the inequality holds true, and graph the solution set on a real number line.

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 positive or negative, which often requires finding the roots of the polynomial and testing intervals between these roots.

Graphing on a Number Line

Graphing the solution set of a polynomial inequality on a number line involves marking the intervals where the inequality holds true. This visual representation helps in understanding the solution's range and is essential for interpreting the results in context, such as identifying where the polynomial is greater than zero.

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 (a, b) includes all numbers between a and b but not a and b themselves, while [a, b] includes both endpoints.