Problem 45 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$2 x^{3}-x^{2}-18 x+9=0$$

Step-by-Step Solution

Answer

The approximate solutions of the given equation are the x-coordinates where the graph of \(f(x) = 2 x^{3}-x^{2}-18 x+9\) crosses the x-axis. These can be found using a graphing utility.

Step 1: Rewrite the Equation in the Form \(f(x) = 0\)

To rewrite the equation \(2 x^{3}-x^{2}-18 x+9 = 0\) in the form \(f(x) = 0\), we simply set \(f(x) = 2 x^{3}-x^{2}-18 x+9\). Therefore, our equation becomes \(f(x) = 0\).

Step 2: Approximate the Solutions Using a Graphing Utility

Next, plot graph of \(f(x) = 2 x^{3}-x^{2}-18 x+9\) using a graphing utility. The x-coordinate of the points where the graph of \(f(x)\) crosses the x-axis are the approximate solutions of the equation.

Key Concepts

Graphing Utility

Graphing utilities are invaluable tools in mathematics, particularly when it comes to visualizing equations and their solutions. A graphing utility, often found on scientific calculators or as software on computers and mobile devices, allows users to input a function and see its graph on a coordinate plane. When working with a cubic equation such as \(2x^{3} - x^{2} - 18x + 9 = 0\), plotting the function can provide a visual representation that makes it easier to identify important features like intercepts, turning points, and the general shape of the graph.

For students, using a graphing utility could involve typing the equation into the program, setting an appropriate scale for the x and y-axes to ensure all relevant points are visible, and then interpreting the resulting graph. It's crucial to understand how to read the graphs produced: the x-intercepts (where the graph crosses the x-axis) indicate the solutions to the equation. By zooming in on these intercept points, students can approximate the values of x for which the function \(f(x) = 0\).

Cubic Equations

Cubic equations, like the one in our example \(2x^{3} - x^{2} - 18x + 9 = 0\), are polynomial equations where the highest exponent of the variable x is three. They can have complex behaviors and up to three real roots or solutions. It's notable that cubic equations always have at least one real root because the graph of a cubic function must cross the x-axis at least once.

Cubic equations create curves with distinctive ‘S’ or ‘∩’ shapes, and typically they turn at most twice, changing direction from increasing to decreasing or vice versa. The general form for a cubic equation is \(ax^{3} + bx^{2} + cx + d = 0\), where a, b, c, and d are constants, and a cannot be zero. Solving cubic equations algebraically can be complex and often involves methods like synthetic division or the rational root theorem. However, finding an approximate solution using a graphing utility is frequently the most accessible approach for students.

X-Intercepts

In the context of graphing, x-intercepts are the points where the graph of a function crosses or touches the x-axis. These points are significant because they represent the solutions to the equation \(f(x) = 0\). When a graph crosses the x-axis, the y-value at that point is zero, which precisely means that the function value \(f(x)\) is zero.

For a cubic equation, there can be up to three x-intercepts, reflecting the fact that a cubic equation can have up to three real solutions. When using a graphing utility, identifying these intercepts visually allows students to approximate their values — though it should be noted that for more accurate solutions, other numerical methods or algebraic manipulation may be required.

• Single x-intercept: Indicates one real root.
• Two or three x-intercepts: Correspond to two or three real roots (solutions).

By carefully analyzing the x-intercepts on the graph, students can better understand the solutions of the cubic equation.