Solving Polynomial Equations: A Step-by-Step Guide

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Solving Polynomial Equations: A Step-by-Step Guide

Hey guys! Let's dive into solving the polynomial equation x(x-2)(x+3) = 18. We're going to tackle this using a graphing calculator and a system of equations. It might sound a bit daunting, but trust me, it's totally doable. So, grab your calculators, and let's get started!

Understanding the Problem

First, let's break down what we're actually trying to do. We have the equation x(x-2)(x+3) = 18, and our mission is to find the value(s) of x that make this equation true. These values are called the roots or solutions of the polynomial equation. Solving polynomial equations can sometimes be tricky, especially when they aren't easily factorable or when dealing with higher-degree polynomials. That's where our trusty graphing calculator comes in handy. It allows us to visualize the equation and find the roots graphically.

Setting up the Equation

Before we jump into using the graphing calculator, let's rearrange the equation to make it equal to zero. This is a standard practice when solving polynomial equations because it allows us to find the x-intercepts, which are the roots. So, we start with:

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

Expanding the left side gives us:

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

x(x^2 + x - 6) = 18

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

Now, subtract 18 from both sides to set the equation to zero:

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

Now we have a polynomial equation in standard form, which is perfect for graphing and finding the roots.

Using a Graphing Calculator

Now comes the fun part – using the graphing calculator to find the roots. If you're new to this, don't worry; I'll walk you through it step by step. We're essentially going to graph the function f(x) = x^3 + x^2 - 6x - 18 and look for the points where the graph crosses the x-axis. These are the real roots of the equation.

Step-by-Step Guide

  1. Turn on your calculator: Make sure your graphing calculator is powered on and ready to go.
  2. Enter the equation: Press the "Y=" button on your calculator. This will allow you to enter the equation. Type in x^3 + x^2 - 6x - 18. You'll usually find the x variable button near the alpha key, and the cube function might be under the math menu or accessible with a caret (^).
  3. Adjust the window: Before graphing, you might need to adjust the window settings to see the graph properly. Press the "WINDOW" button. You'll see options like Xmin, Xmax, Ymin, and Ymax. A good starting point might be setting Xmin to -5, Xmax to 5, Ymin to -20, and Ymax to 20. This gives you a decent view of the graph around the origin. Adjust as needed based on what you see.
  4. Graph the equation: Press the "GRAPH" button. You should now see the graph of the polynomial function. Look for the points where the graph intersects the x-axis.
  5. Find the roots: Use the calculator's built-in functions to find the x-intercepts (roots). Press "2nd" then "TRACE" (which is the CALC button). Choose option 2: "zero".
  6. Follow the prompts: The calculator will ask you for a "Left Bound," "Right Bound," and a "Guess." For the left bound, move the cursor to the left of the x-intercept you want to find and press "ENTER." For the right bound, move the cursor to the right of the x-intercept and press "ENTER." For the guess, move the cursor close to the x-intercept and press "ENTER."
  7. Read the root: The calculator will display the x-value of the root. This is the solution to your equation. Repeat the process for any other x-intercepts you see on the graph.

Analyzing the Graph

When you graph the equation f(x) = x^3 + x^2 - 6x - 18, you'll notice that it crosses the x-axis at only one point. This indicates that there is only one real root for this equation. By using the calculator's zero function, you'll find that this root is approximately x = 3. This means that x = 3 is a solution to the equation x(x-2)(x+3) = 18.

Verifying the Solution

To make sure our solution is correct, let's plug x = 3 back into the original equation:

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

3(1)(6) = 18

18 = 18

Yep, it checks out! So, x = 3 is indeed a root of the polynomial equation.

Setting up a System of Equations

Now, let's explore how we can approach this problem using a system of equations. This method might seem a bit unconventional for this specific problem, but it’s a great way to understand different problem-solving techniques. The idea here is to introduce new variables to break down the original equation into smaller, more manageable parts.

Breaking Down the Equation

We start with our original equation:

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

Let's introduce two new variables, y and z, such that:

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

Now our equation becomes:

y * z = 18*

So we have the following system of equations:

  1. y = x(x-2)
  2. z = x+3
  3. y * z = 18*

Solving the System

This system is a bit tricky to solve directly using algebraic methods, but we can use the graphing calculator to help us visualize and find potential solutions. Here’s how we can do it:

  1. Express everything in terms of x: We already have y and z expressed in terms of x. So, we have:

    • y = x^2 - 2x
    • z = x + 3

  2. Substitute into the third equation: Substitute y and z into the third equation:

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

    This gives us the same polynomial equation we had before:

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

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

  3. Graph the equation: As we did before, graph f(x) = x^3 + x^2 - 6x - 18 using the graphing calculator. Find the x-intercept, which will give us the value of x that satisfies the equation.

  4. Verify the solution: Once you find a potential solution for x, plug it back into the original equations to make sure it holds true.

Why This Method?

While setting up a system of equations might seem like an overkill for this particular problem, it illustrates a useful technique for more complex problems. Breaking down a complex equation into smaller parts can make it easier to analyze and solve, especially when dealing with multiple variables or constraints. In many engineering and scientific applications, systems of equations are commonly used to model and solve complex problems.

Conclusion

So, there you have it! We've successfully found the root of the polynomial equation x(x-2)(x+3) = 18 using both a graphing calculator and a system of equations. The root is x = 3. Remember, the graphing calculator is a powerful tool for visualizing equations and finding solutions, while setting up systems of equations can help break down complex problems into more manageable parts. Keep practicing, and you'll become a pro at solving polynomial equations in no time! Keep your chin up and keep trying and good luck!