Use the graph to solve each equation or inequality. Use interval notation where appropriate. 7x(x - 1)(x - 2) = 0

Verified step by step guidance

1. Identify the roots of the equation by setting each factor equal to zero: 7x = 0, x - 1 = 0, and x - 2 = 0.
2. Solve each equation to find the roots: x = 0, x = 1, and x = 2.
3. These roots represent the x-values where the graph intersects the x-axis.
4. To solve the inequality 7x(x - 1)(x - 2) = 0, observe the intervals where the graph is above or below the x-axis.
5. Use interval notation to express the solution set for the inequality based on the intervals where the graph is above or below the x-axis.

Key Concepts

Roots of a Polynomial

The roots of a polynomial are the values of x for which the polynomial equals zero. In the equation 7x(x - 1)(x - 2) = 0, the roots can be found by setting each factor to zero, resulting in x = 0, x = 1, and x = 2. These roots correspond to the x-intercepts of the graph, where the curve crosses the x-axis.

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 interval) or excluded (open interval). For example, the interval [0, 2] includes both endpoints, while (0, 2) does not. This notation is essential for expressing solutions to inequalities derived from polynomial equations.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting points based on the function's values and identifying key features such as roots, turning points, and end behavior. The graph of the polynomial 7x(x - 1)(x - 2) will show the roots at x = 0, x = 1, and x = 2, and the behavior of the graph in the intervals between and beyond these roots is crucial for solving inequalities and understanding the function's overall shape.