Finding Roots Of Polynomial Equation X³ - 5x + 5 = 2x² - 5 Using Graphing Calculator

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Finding the roots of polynomial equations can sometimes be a challenging task, especially when dealing with cubic or higher-degree polynomials. In this comprehensive guide, we will delve into the process of determining the roots of the polynomial equation x³ - 5x + 5 = 2x² - 5. We will employ a graphing calculator and a system of equations to solve this problem, providing a step-by-step approach to help you understand the underlying concepts and techniques. This method allows us to visualize the solutions and approximate non-integer roots with precision.

Rewriting the Equation

Before we begin using the graphing calculator, it's essential to rewrite the given equation in a standard form. This will make it easier to input the equation into the calculator and interpret the results. Our initial equation is:

x³ - 5x + 5 = 2x² - 5

To rewrite this in standard form, we need to move all terms to one side of the equation, setting it equal to zero. This is achieved by subtracting 2x² and adding 5 to both sides of the equation:

x³ - 2x² - 5x + 10 = 0

Now, we have a cubic polynomial equation in the standard form, which is ready for further analysis using a graphing calculator. This form, x³ - 2x² - 5x + 10 = 0, allows us to easily identify the coefficients and the constant term, which are crucial for both graphical and analytical methods of solving the equation. Having the equation in this form simplifies the process of finding the roots, as it aligns with the standard representation of polynomial equations.

Using a Graphing Calculator

The graphing calculator is an invaluable tool for finding the roots of polynomial equations. It allows us to visualize the function and identify the points where the graph intersects the x-axis, which represent the real roots of the equation. To use the graphing calculator effectively, we will follow these steps:

  1. Enter the Equation: First, we need to input the rewritten equation into the calculator. Access the equation editor (usually denoted as Y=) and enter the polynomial function y = x³ - 2x² - 5x + 10. Ensure that the equation is entered correctly to avoid any errors in the graph.
  2. Set the Viewing Window: The viewing window determines the portion of the coordinate plane that is displayed on the calculator screen. It's important to set an appropriate window to see the relevant parts of the graph. A standard window setting might be a good starting point, but you may need to adjust it to better visualize the roots. For this equation, a window that includes x-values from -5 to 5 and y-values from -10 to 20 should provide a clear view of the graph and its intersections with the x-axis.
  3. Graph the Function: Once the equation is entered and the viewing window is set, graph the function. Observe the points where the graph crosses the x-axis. These points represent the real roots of the equation. The calculator will display the curve of the polynomial, allowing you to see its behavior and identify potential roots visually.
  4. Find the Roots: Use the calculator's built-in function to find the roots (also known as zeros or x-intercepts). This function is typically found under the “CALC” menu (usually accessed by pressing the second function key and the trace key). Select the “zero” option and follow the prompts to select the left bound, right bound, and a guess near the root you want to find. The calculator will then approximate the root with high precision. Repeat this process for each intersection point to find all real roots of the equation.

By following these steps, the graphing calculator will help us identify both integer and non-integer roots of the polynomial equation. The visual representation of the graph also provides a deeper understanding of the function's behavior and its relationship to the roots.

Finding Integer Roots

From the graph, we can observe the points where the curve intersects the x-axis. These intersections represent the real roots of the equation x³ - 2x² - 5x + 10 = 0. Integer roots are those intersections that occur at integer values of x. By examining the graph generated by the graphing calculator, we can identify one clear integer root. The graph appears to cross the x-axis at x = 2. To confirm this, we can substitute x = 2 into the equation:

(2)³ - 2(2)² - 5(2) + 10 = 8 - 8 - 10 + 10 = 0

Since the equation equals zero when x = 2, we can confirm that x = 2 is indeed an integer root of the polynomial equation. This verification step is crucial to ensure the accuracy of the roots identified from the graph. Integer roots are often easier to identify and confirm, making them a valuable starting point in the process of solving polynomial equations. The ability to find integer roots can simplify the subsequent steps, such as polynomial division, to find the remaining roots.

Using Synthetic Division

Now that we've identified one integer root, x = 2, we can use synthetic division to factor the polynomial and find the remaining roots. Synthetic division is an efficient method for dividing a polynomial by a linear factor. To perform synthetic division, we set up the coefficients of the polynomial and the root we are dividing by. The polynomial is x³ - 2x² - 5x + 10, so the coefficients are 1, -2, -5, and 10. We'll use the root 2 for the synthetic division.

Here are the steps for synthetic division:

  1. Write down the coefficients of the polynomial: 1, -2, -5, 10.
  2. Write the root (2) to the left.
  3. Bring down the first coefficient (1) to the bottom row.
  4. Multiply the root (2) by the first number in the bottom row (1) and write the result (2) under the second coefficient (-2).
  5. Add the numbers in the second column (-2 + 2 = 0) and write the result (0) in the bottom row.
  6. Multiply the root (2) by the second number in the bottom row (0) and write the result (0) under the third coefficient (-5).
  7. Add the numbers in the third column (-5 + 0 = -5) and write the result (-5) in the bottom row.
  8. Multiply the root (2) by the third number in the bottom row (-5) and write the result (-10) under the fourth coefficient (10).
  9. Add the numbers in the fourth column (10 + (-10) = 0) and write the result (0) in the bottom row. This is the remainder.

The result of the synthetic division gives us the coefficients of the quotient polynomial. The numbers in the bottom row (excluding the remainder) are 1, 0, and -5. This corresponds to the quadratic polynomial x² + 0x - 5, which simplifies to x² - 5. The zero remainder confirms that 2 is indeed a root, and (x - 2) is a factor of the original polynomial.

Solving the Quadratic Equation

After performing synthetic division, we obtained the quadratic equation x² - 5 = 0. To find the remaining roots, we need to solve this equation. This can be done by isolating x² and then taking the square root of both sides:

x² = 5

Taking the square root of both sides, we get:

x = ±√5

This gives us two additional roots: x = √5 and x = -√5. These are non-integer roots, and we can approximate their values using a calculator. The square root of 5 is approximately 2.236, so the roots are approximately x ≈ 2.24 and x ≈ -2.24 when rounded to the nearest hundredth. Solving the quadratic equation provides the remaining roots of the original polynomial, completing the process of finding all the solutions.

Final Roots

In summary, we have found all the roots of the polynomial equation x³ - 5x + 5 = 2x² - 5 using a combination of graphing calculator analysis, synthetic division, and algebraic methods. The roots are:

  1. x = 2 (integer root)
  2. x ≈ 2.24 (non-integer root, rounded to the nearest hundredth)
  3. x ≈ -2.24 (non-integer root, rounded to the nearest hundredth)

These roots represent the values of x for which the polynomial equation equals zero. The graphing calculator provided a visual representation of the roots, while synthetic division and solving the quadratic equation allowed us to find the exact and approximate values of the roots. Understanding these methods is crucial for solving polynomial equations of various degrees.

By utilizing these techniques, we can effectively find the roots of polynomial equations, whether they are integer or non-integer values. This comprehensive approach ensures accuracy and provides a deeper understanding of the solutions and the behavior of polynomial functions.