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. First, rewrite the inequality clearly: \(x^3 + x^2 + 4x + 4 > 0\).
2. Next, factor the cubic polynomial if possible. Try to factor by grouping: group terms as \((x^3 + x^2) + (4x + 4)\).
3. Factor out common factors from each group: \(x^2(x + 1) + 4(x + 1)\).
4. Since both groups contain the factor \((x + 1)\), factor it out: \((x + 1)(x^2 + 4) > 0\).
5. Analyze the factors separately: \(x + 1\) and \(x^2 + 4\). Note that \(x^2 + 4\) is always positive for all real \(x\) because \(x^2 \\geq 0\) and adding 4 keeps it positive. So the inequality depends on the sign of \(x + 1\).

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 breaking down a polynomial into simpler polynomials (factors) whose product equals the original polynomial. This helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign.

Interval Notation and Number Line Graphing

Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial is positive or negative, helping to interpret and communicate the solution clearly.