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. x³≤4x²

Verified step by step guidance

1. Rewrite the inequality \(x^{3} \leq 4x^{2}\) by bringing all terms to one side to set the inequality to zero: \(x^{3} - 4x^{2} \leq 0\).
2. Factor the left-hand side expression: first, factor out the greatest common factor \(x^{2}\) to get \(x^{2}(x - 4) \leq 0\).
3. Identify the critical points by setting each factor equal to zero: \(x^{2} = 0\) gives \(x = 0\), and \(x - 4 = 0\) gives \(x = 4\). These points divide the real number line into intervals to test.
4. Test the sign of the expression \(x^{2}(x - 4)\) in each interval determined by the critical points: \((-\infty, 0)\), \((0, 4)\), and \((4, \infty)\), to determine where the inequality holds true.
5. Based on the sign test, write the solution set where \(x^{2}(x - 4) \leq 0\) is true, including points where the expression equals zero, and express the solution in interval notation.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another expression 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 rewriting a polynomial as a product of simpler polynomials or factors. It helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign in inequalities.

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 inequality holds true, using open or closed dots to indicate whether endpoints are included.