Finding Roots: Polynomial Equation With Calculator

Hey guys! Today, we're diving into how to find the roots of the polynomial equation $x^4 + x^2 = 4x^3 - 12x + 12$. We'll be using a graphing calculator and exploring a system of equations to crack this problem. Get ready, and let's make math a little less intimidating!

Understanding Polynomial Roots

Before we jump into solving, let's quickly recap what polynomial roots are. Essentially, the roots of a polynomial equation are the values of x that make the equation equal to zero. These roots are also the x-intercepts of the polynomial function when graphed. Finding these roots can sometimes be straightforward, but for higher-degree polynomials like the one we're tackling today, it often requires some handy tools like graphing calculators.

Roots, also known as zeros or solutions, play a crucial role in understanding the behavior of polynomial functions. They help in sketching the graph of the function, determining intervals where the function is positive or negative, and solving real-world problems modeled by polynomials. When dealing with polynomials, especially those of higher degrees, finding roots becomes an essential skill in mathematical analysis.

For instance, consider a simple quadratic equation $x^2 - 5x + 6 = 0$. By factoring, we can find that $(x - 2)(x - 3) = 0$, which gives us the roots $x = 2$ and $x = 3$. These roots tell us where the parabola intersects the x-axis. Similarly, for more complex polynomials, the roots provide valuable information about the function's behavior. In many practical applications, roots can represent critical values or thresholds in the system being modeled. Understanding how to find these roots, whether through algebraic methods or numerical tools, is fundamental in various fields of science, engineering, and economics.

Setting Up the Equation

First, we need to rearrange the given equation into a standard polynomial form. This means setting the equation equal to zero. So, let's rewrite $x^4 + x^2 = 4x^3 - 12x + 12$ as:

$x^4 - 4x^3 + x^2 + 12x - 12 = 0$

Now we have a polynomial equation of degree 4, which can be a bit tricky to solve directly. That's where our graphing calculator comes to the rescue!

Why Graphing Calculators are Super Helpful

Graphing calculators are not just for plotting graphs; they are powerful tools for solving equations, especially polynomial equations that are difficult to solve algebraically. They can quickly find the roots (x-intercepts) of a polynomial, which are the solutions to the equation when it is set to zero. Graphing calculators use numerical methods to approximate the roots, making them indispensable for equations without simple, rational solutions.

Moreover, graphing calculators allow us to visualize the behavior of the polynomial function. By observing the graph, we can estimate the number and approximate values of the roots. This visual representation can guide us in further analysis or in setting up systems of equations to find the roots more accurately. Graphing calculators also provide functionalities like zooming in on specific regions of the graph, which can help in refining the approximation of the roots.

In summary, a graphing calculator can transform the daunting task of finding polynomial roots into a more manageable and intuitive process. It reduces the reliance on complex algebraic manipulations and provides a visual confirmation of the solutions, making it an invaluable tool for students and professionals alike.

Using the Graphing Calculator

1. Enter the Equation: Input the polynomial $y = x^4 - 4x^3 + x^2 + 12x - 12$ into your graphing calculator.
2. Plot the Graph: Plot the graph to visualize the function. You should see where the graph intersects the x-axis. These intersection points are the real roots of the equation.
3. Find the Roots: Use the calculator's built-in function to find the zeros (roots) of the equation. This usually involves using the "zero," "root," or "intersect" function.
4. Approximate the Roots: The calculator will give you the approximate values of the roots. Round these values to the nearest hundredth as required.

Step-by-Step Instructions for TI-84 Calculator

For those of you using a TI-84 calculator, here’s a more detailed breakdown:

1. Press Y= and enter the equation $x^4 - 4x^3 + x^2 + 12x - 12$ into Y1.
2. Press GRAPH to plot the function. If the graph is not clearly visible, adjust the window settings by pressing WINDOW and setting appropriate values for Xmin, Xmax, Ymin, and Ymax.
3. To find the roots, press 2ND then TRACE (CALC menu). Choose option 2: zero.
4. The calculator will ask for a "Left Bound." Use the arrow keys to move the cursor to the left of the root you want to find and press ENTER.
5. Next, it will ask for a "Right Bound." Move the cursor to the right of the root and press ENTER.
6. Finally, it will ask for a "Guess." Move the cursor close to the root and press ENTER.
7. The calculator will display the approximate value of the root.
8. Repeat these steps for each root of the equation.

Analyzing the Graph and Roots

From the graph, you should observe that the polynomial intersects the x-axis at a few points. These points represent the real roots of the equation. Using the calculator, you'll find that the roots are approximately:

• x ≈ -1.73
• x ≈ 1.73
• x = 2

So, we have three real roots: approximately -1.73, 1.73, and 2. Notice that the polynomial touches the x-axis at x = 2 but doesn't cross it, indicating that x = 2 might be a repeated root. Let's verify this using polynomial division or synthetic division.

Confirming the Repeated Root

To confirm that x = 2 is a repeated root, we can perform synthetic division on the polynomial $x^4 - 4x^3 + x^2 + 12x - 12$ using x = 2. If 2 is indeed a repeated root, the quotient polynomial should also have 2 as a root.

Performing synthetic division once gives us:

2 | 1  -4   1  12  -12
  |     2  -4  -6   12
  ----------------------
    1  -2  -3   6    0

The resulting polynomial is $x^3 - 2x^2 - 3x + 6$. Since the remainder is 0, x = 2 is a root. Now, let's perform synthetic division again on the new polynomial:

2 | 1  -2  -3   6
  |     2   0  -6
  ----------------
    1   0  -3   0

The resulting polynomial is $x^2 - 3$, and the remainder is again 0. This confirms that x = 2 is a repeated root. Now we can easily solve $x^2 - 3 = 0$ to find the remaining roots.

Solving the Remaining Quadratic

After confirming that x = 2 is a repeated root, we're left with the quadratic equation $x^2 - 3 = 0$. Solving this equation is straightforward:

$x^2 - 3 = 0$

$x^2 = 3$

$x = ±√3$

So, the roots are $x = √3$ and $x = -√3$. Approximating these to the nearest hundredth, we get:

$x ≈ 1.73$

$x ≈ -1.73$

Final Roots and Answer

Therefore, the roots of the polynomial equation $x^4 + x^2 = 4x^3 - 12x + 12$ are:

• x ≈ -1.73
• x ≈ 1.73
• x = 2 (repeated root)

Looking at the multiple-choice options, the correct answer is:

C. -1.73, 1.73, 2

Hope that helps, and happy calculating!