What Is The Root Of The Polynomial Equation X(x-2)(x+3)=18? Solve Using A Graphing Calculator And A System Of Equations.

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In the realm of mathematics, specifically in algebra, finding the roots of polynomial equations is a fundamental task. Roots, also known as solutions or zeros, are the values of the variable that make the equation true. One common type of equation encountered is the polynomial equation, where the highest power of the variable is greater than one. Finding the roots of a polynomial equation can be challenging, especially for higher-degree polynomials. In this comprehensive guide, we will explore a graphical approach to determine the roots of polynomial equations, focusing on using a graphing calculator and the concept of a system of equations. This method provides a visual and intuitive way to understand the solutions.

Understanding Polynomial Equations

To effectively find the roots of a polynomial equation, it's crucial to first understand what a polynomial equation is. A polynomial equation is an equation that involves variables raised to non-negative integer powers. The general form of a polynomial equation is:

a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0

where:

  • x is the variable.
  • n is a non-negative integer representing the degree of the polynomial.
  • a_n, a_{n-1}, ..., a_1, a_0 are constant coefficients.

Roots or solutions of a polynomial equation are the values of x that satisfy the equation, meaning when these values are substituted into the equation, the left-hand side equals zero. Graphically, the roots correspond to the points where the graph of the polynomial function intersects the x-axis. These intersections are also known as x-intercepts.

The Power of Graphing Calculators

Graphing calculators are powerful tools that can visually represent mathematical functions, including polynomials. By graphing a polynomial function, we can easily identify its roots as the points where the graph crosses the x-axis. This method is particularly useful for higher-degree polynomials where algebraic methods can become complex and time-consuming.

To use a graphing calculator for finding roots:

  1. Enter the Polynomial: Input the polynomial equation into the calculator's function editor (usually the "Y=" menu).
  2. Graph the Function: Plot the graph of the polynomial function.
  3. Identify Intercepts: Look for the points where the graph intersects the x-axis. These points are the real roots of the equation.
  4. Use Calculator Tools: Most graphing calculators have built-in tools like "zero" or "root" finders that can accurately calculate the x-intercepts.

Solving Polynomial Equations as a System of Equations

An alternative approach to finding the roots of a polynomial equation using a graphing calculator is to treat it as a system of equations. This method involves breaking down the polynomial equation into two simpler equations and finding their intersection points.

For example, consider the given polynomial equation:

x(x-2)(x+3) = 18

We can rewrite this equation by introducing a new variable y and forming the following system of equations:

y = x(x-2)(x+3)
y = 18

The roots of the original polynomial equation are the x-coordinates of the points where the graphs of these two equations intersect. Graphing these equations on a graphing calculator allows us to visually identify these intersection points and determine the roots.

Step-by-Step Solution

Let's apply the system of equations method to solve the given polynomial equation:

x(x-2)(x+3) = 18

  1. Rewrite as a System of Equations

    Define two functions:

    y_1 = x(x-2)(x+3)
y_2 = 18

  2. Enter into Graphing Calculator

    Input these equations into the graphing calculator. In the "Y=" menu, enter y_1 as X(X-2)(X+3) and y_2 as 18.

  3. Graph the Equations

    Graph both equations on the same coordinate plane. Adjust the viewing window if necessary to see the intersection points clearly. A standard window might not be sufficient, so you may need to zoom out or adjust the x and y ranges.

  4. Identify Intersection Points

    Look for the points where the graphs of y_1 and y_2 intersect. These points represent the solutions to the system of equations and, therefore, the roots of the original polynomial equation.

  5. Use Intersection Tool

    Most graphing calculators have an "intersect" function that can accurately find the coordinates of the intersection points. Use this tool to determine the x-coordinates, which are the roots of the polynomial.

  6. Analyze the Graph

    The graph of y_1 = x(x-2)(x+3) is a cubic polynomial, and the graph of y_2 = 18 is a horizontal line. By graphing these, we can see where the polynomial function's value is equal to 18. This graphical representation provides a clear visualization of the solutions.

Detailed Graphical Analysis

To further illustrate the process, let’s perform a more detailed analysis of the graphical solution. First, we expand the polynomial equation to its standard form:

x(x-2)(x+3) = x(x^2 + x - 6) = x^3 + x^2 - 6x

So, the equation becomes:

x^3 + x^2 - 6x = 18

Rearranging the equation, we get:

x^3 + x^2 - 6x - 18 = 0

Now, we can define our two functions:

y_1 = x^3 + x^2 - 6x
y_2 = 18

Graphing these functions using a graphing calculator, we can observe the following:

  • The cubic function y_1 = x^3 + x^2 - 6x has a curve that intersects the x-axis at three points, representing the roots of the polynomial x^3 + x^2 - 6x = 0. However, we are interested in the points where y_1 = 18.
  • The horizontal line y_2 = 18 represents the constant value 18. We are looking for the x-values where the cubic function's graph intersects this horizontal line.

By adjusting the viewing window on the graphing calculator, we can zoom in on the intersection points. Typically, a window setting of -5 ≤ x ≤ 5 and -20 ≤ y ≤ 30 will provide a clear view of the relevant part of the graph.

Using the calculator's intersect function, we find one real intersection point at approximately x = 3. This means that x = 3 is a root of the equation x(x-2)(x+3) = 18.

Verification and Discussion of Possible Roots

To verify that x = 3 is indeed a root, we substitute it back into the original equation:

3(3-2)(3+3) = 3(1)(6) = 18

Since the equation holds true, x = 3 is a valid root.

Now, let's consider the provided options:

A. -3 B. 0 C. 2 D. 3

We have already verified that option D, x = 3, is a root. Let's test the other options:

A. For x = -3:

(-3)(-3-2)(-3+3) = (-3)(-5)(0) = 0 ≠ 18

B. For x = 0:

0(0-2)(0+3) = 0 ≠ 18

C. For x = 2:

2(2-2)(2+3) = 2(0)(5) = 0 ≠ 18

Thus, only x = 3 satisfies the equation.

Advantages of the Graphical Method

Using a graphing calculator and the system of equations approach offers several advantages:

  1. Visual Representation: Graphs provide a visual understanding of the solutions, making it easier to comprehend the concept of roots.
  2. Complex Equations: This method is particularly useful for higher-degree polynomials or equations that are difficult to solve algebraically.
  3. Real Roots: Graphing calculators can efficiently find real roots, which are the points where the graph intersects the x-axis.
  4. Estimation: Even if exact solutions are not immediately apparent, the graph allows for a good estimation of the roots.

Limitations and Considerations

While the graphical method is powerful, it has certain limitations:

  1. Accuracy: The accuracy of the roots found depends on the precision of the graphing calculator and the viewing window settings. Sometimes, it might be necessary to use algebraic methods for exact solutions.
  2. Imaginary Roots: The graphical method primarily identifies real roots. Complex or imaginary roots, which do not appear as x-intercepts, are not directly visible on the graph.
  3. Time Consuming: Although it provides a visual aid, setting up and analyzing the graph can sometimes be time-consuming, especially if adjustments to the viewing window are frequently required.

Alternative Methods

Besides the graphical method, there are other ways to find the roots of polynomial equations:

  1. Factoring: If the polynomial can be factored, the roots can be found by setting each factor equal to zero.
  2. Rational Root Theorem: This theorem helps identify potential rational roots of a polynomial equation.
  3. Synthetic Division: Synthetic division is a method to test potential rational roots and reduce the degree of the polynomial.
  4. Numerical Methods: Methods like Newton's method can approximate the roots of a polynomial equation.
  5. Algebraic Formulas: For quadratic equations, the quadratic formula provides a direct way to find the roots.

Conclusion

Finding the roots of polynomial equations is a crucial skill in mathematics. The graphical method, using a graphing calculator and the system of equations approach, provides an effective and intuitive way to solve these equations. By graphing the polynomial function and identifying the x-intercepts or the intersection points of the system of equations, we can determine the real roots. While this method has its limitations, it is particularly useful for visualizing and estimating the solutions of complex polynomial equations. By combining the graphical method with other algebraic techniques, we can comprehensively address the problem of finding roots of polynomial equations.

In the specific case of the equation x(x-2)(x+3) = 18, the graphical method clearly demonstrates that x = 3 is the correct root. This comprehensive guide has illustrated the power and versatility of the graphical approach in solving polynomial equations, emphasizing its importance in mathematical problem-solving.