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 polynomial from the equation \$7x(x - 1)(x - 2) = 0\(. These roots are the values of \)x\( that make the equation equal to zero. Set each factor equal to zero: \)7x = 0\(, \)x - 1 = 0\(, and \)x - 2 = 0$.
2. Solve each equation to find the roots: \(x = 0\), \(x = 1\), and \(x = 2\). These are the points where the graph crosses the x-axis.
3. To solve the equation \$7x(x - 1)(x - 2) = 0\(, recognize that the solutions are exactly the roots found: \)x = 0\(, \)x = 1\(, and \)x = 2$.
4. To solve inequalities such as \$7x(x - 1)(x - 2) > 0\( or \)7x(x - 1)(x - 2) < 0\(, analyze the sign of the polynomial in the intervals determined by the roots: \)(-\infty, 0)\(, \)(0, 1)\(, \)(1, 2)\(, and \)(2, \infty)$.
5. Use the graph to determine where the polynomial is positive or negative by observing whether the graph is above or below the x-axis in each interval. Then express the solution to the inequality using interval notation based on these observations.

Key Concepts

Roots of a Polynomial

Roots or zeros of a polynomial are the values of x where the polynomial equals zero. For the equation 7x(x - 1)(x - 2) = 0, the roots are x = 0, x = 1, and x = 2. These points correspond to where the graph intersects the x-axis.

Solving Polynomial Inequalities Using Graphs

To solve inequalities like 7x(x - 1)(x - 2) > 0 or < 0, analyze the graph to determine where the polynomial is above or below the x-axis. Intervals where the graph is above the x-axis correspond to positive values, and intervals below correspond to negative values.

Interval Notation

Interval notation is a concise way to express sets of numbers between two endpoints. For polynomial inequalities, intervals between roots where the function is positive or negative are written using parentheses or brackets, indicating whether endpoints are included or excluded.