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≤4x^2

Verified step by step guidance

1. Start by moving all terms to one side of the inequality to set it to zero: $x^3-4x^2\le 0$.
2. Factor the polynomial on the left-hand side: $x^2(x-4)\le 0$.
3. Identify the critical points by setting each factor equal to zero: $x^2=0$ and $x-4=0$, giving critical points $x=0$ and $x=4$.
4. Use the critical points to divide the number line into intervals: $(-\mathrm{\infty },0)$, $(0,4)$, and $(4,\mathrm{\infty })$.
5. Test a point from each interval in the inequality $x^2(x-4)\le 0$ to determine where the inequality holds true, and then express the solution set in interval notation.

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 finds the roots of the corresponding polynomial equation and tests intervals between these roots to determine where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). For example, the interval [a, b) includes 'a' but not 'b', while (a, b) excludes both endpoints.

Graphing Solutions on a Number Line

Graphing solutions on a number line visually represents the solution set of an inequality. Each solution is marked with a dot or a line segment, depending on whether the endpoints are included or excluded. This graphical representation helps in understanding the range of values that satisfy the inequality.