Solving Polynomial Equations: Graphing & Systems

Hey math enthusiasts! Today, we're diving into a classic problem: figuring out the root of the polynomial equation x(x-2)(x+3) = 18. You might be thinking, "Ugh, polynomials!" But trust me, we'll break it down in a super approachable way, using tools like a graphing calculator and systems of equations to make the process a breeze. Let's get started, guys!

Understanding the Problem: The Root of a Polynomial

So, what exactly are we trying to find? In simple terms, the "root" (also known as the zero or the solution) of a polynomial equation is the value(s) of x that make the equation true. In our case, we need to find the x value(s) that satisfy x(x-2)(x+3) = 18. This means we're looking for the points where the graph of the polynomial function intersects with the horizontal line y = 18. Essentially, we're trying to figure out where the function's output equals 18. This might seem a little abstract, but it'll become clearer as we move through the process.

One thing that is important is that polynomial equations are used to model so many real-world phenomena. From physics, where they can describe the motion of a projectile, to economics, where they can model cost or revenue curves, understanding polynomials gives you a powerful tool. And finding their roots is a fundamental skill in these applications! Think of it like this: finding the root allows us to pinpoint specific points in a system that we're trying to understand. It's like finding the key to unlock a problem. So, are you ready to jump into the different methods? Because we're about to make solving polynomials easy.

First, we'll look at the graphical method. This is where we will plot our equation on a graphing calculator, because it will give us a very clear visual representation of what's going on, and allows us to visually see the solutions. Then, we will look at how we can use a system of equations, a more algebraic method, to get the same answer in a slightly different way. Each method gives a unique perspective, and it's always a good idea to know how to solve a problem with different tools!

Remember, the goal is always to find the values of x that make the original equation true. The root is a number. In this problem, we're going to use several methods to solve the problem and determine the answer, which we already know is one of the multiple-choice options, which makes things easier! And it's important to remember that some polynomial equations might have multiple roots, a single root, or, in some cases, no real roots at all. But don't worry, we'll take it one step at a time, using our trusty tools to nail down the correct answer.

Method 1: Using a Graphing Calculator

Alright, let's get our hands dirty with the first method: using a graphing calculator. This is one of the most visual and intuitive ways to tackle this problem, guys.

Here’s how we'll do it. First, we need to rewrite our equation slightly to make it easier to graph. We can rearrange x(x-2)(x+3) = 18 into x(x-2)(x+3) - 18 = 0. Now, you can enter the left side of the equation, y = x(x-2)(x+3) - 18, into your graphing calculator. Make sure your calculator is in function mode. When you graph this, you'll see a curve (a cubic polynomial, to be exact) on your screen. The roots are the x-intercepts, where the graph crosses the x-axis (where y = 0). But that's not what we're looking for! Remember, we need to find the points where our original equation equals 18. So, the best way to do that is to graph two equations: y1 = x(x-2)(x+3) and y2 = 18. Then, we need to find the x-coordinate(s) of the point(s) where these two graphs intersect. This is where the output of the function, x(x-2)(x+3), equals 18.

Use your calculator's "intersect" function. Most graphing calculators have this feature, usually found in the "calc" or "graph" menu. After selecting the intersect option, the calculator will prompt you to select the two curves and provide a guess near the point of intersection. Just follow the instructions on your calculator; it usually involves moving the cursor near the intersection point and pressing enter a few times. The calculator will then display the coordinates of the intersection point. The x-coordinate of that intersection point is the root of our equation.

By carefully examining the graph and using the "intersect" function, we will find that the graph of y = x(x-2)(x+3) intersects with the line y = 18 at x = 3. You might find other intersections. However, since the prompt only gives one answer, this must be the solution to the problem. Congratulations, we've found our first solution: x = 3! This graphical method is super helpful because it gives you a visual understanding of the solution. You can immediately see the point where the equation equals 18. Also, it’s a great way to check your work when you’re solving equations using other methods.

Now, let's move on to another method for an additional perspective on solving this type of equation. It’s always good to use a second method to double-check that your work is accurate. Remember, the key is to understand the problem from multiple angles and to find the method that works best for you. Let's move on!

Method 2: Solving with a System of Equations

Now, let's shift gears and solve this equation using a system of equations. This method offers a more algebraic approach to the problem. We want to find a way to express our polynomial equation as a system of two separate equations, and then use algebra to find the intersection of the two. This can give us an alternative approach, and help make sure we have the correct answer.

First, let's rewrite the original equation, x(x-2)(x+3) = 18. Instead of dealing with the expanded form of this cubic polynomial directly, let's break it down into two simpler equations. We can set up the system as follows:

• Equation 1: y = x(x-2)(x+3)
• Equation 2: y = 18

This system represents the same problem, but now we're looking for the points (x, y) where these two equations intersect. You can think of it like this: Equation 1 is our polynomial function, and Equation 2 is a horizontal line at y = 18. Where these two lines intersect, the value of the polynomial equals 18. Remember that solving a system of equations means finding the values of the variables that satisfy all equations simultaneously. In this case, we're solving for x and y that work in both Equation 1 and Equation 2.

Now, substitute the value of y from Equation 2 into Equation 1. We know y = 18, so replace y in the first equation. This gives us:

• 18 = x(x-2)(x+3)

This is the same as the original equation. But now, we are set up to verify our solution by using the graph. It also allows us to see how we could have used algebra to manipulate the equation and solve for x.

To solve for x algebraically, you would need to expand the polynomial, set the equation equal to zero, and then attempt to factor it or use other methods such as the quadratic formula. However, since we're using the answers to the question, we can substitute each answer choice into the original equation, x(x-2)(x+3) = 18, and see which one makes the equation true. Let's try the answer choices:

• A. -3: (-3)(-3-2)(-3+3) = 0 ≠ 18
• B. 0: (0)(0-2)(0+3) = 0 ≠ 18
• C. 2: (2)(2-2)(2+3) = 0 ≠ 18
• D. 3: (3)(3-2)(3+3) = 18 = 18

Answer choice D, x = 3, makes the equation true. Therefore, the solution to the polynomial equation x(x-2)(x+3) = 18 is x = 3.

And there you have it, guys! We've successfully solved the polynomial equation using both graphical and algebraic methods. Remember that understanding the roots of polynomial equations is a fundamental skill in math. It unlocks the ability to model and solve complex real-world problems. Keep practicing, and you'll become a pro at these problems in no time. Good luck, and happy solving!