Finding Roots Of Polynomial Equation $x^3 - 10x = -3x^2 + 24$

In the realm of mathematics, polynomial equations hold a significant position, appearing in various fields such as algebra, calculus, and engineering. Solving polynomial equations involves finding the values of the variable that satisfy the equation, which are also known as the roots or zeros of the polynomial. There are various methods to find the roots of polynomial equations, including factoring, using the quadratic formula, and employing numerical methods such as the Newton-Raphson method. However, for higher-degree polynomials, these methods can become cumbersome and time-consuming. In such cases, graphing calculators and systems of equations offer an efficient and visual approach to determine the roots.

One particularly effective technique involves utilizing graphing calculators and systems of equations. This method allows for a visual representation of the polynomial equation, making it easier to identify the roots. By graphing the polynomial, we can visually locate the points where the graph intersects the x-axis, which represent the real roots of the equation. Furthermore, we can rewrite the polynomial equation as a system of equations, which can then be solved using various algebraic techniques or with the aid of a calculator. This approach is especially useful for higher-degree polynomials where traditional methods may be more complex. Let's delve into the process of solving the polynomial equation $x^3 - 10x = -3x^2 + 24$ using a graphing calculator and a system of equations, to illustrate the power and efficiency of these techniques in finding the roots of polynomial equations.

Before we dive into the solution, let's first understand what a polynomial equation is. A polynomial equation is an equation that can be written in the form:

$a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 = 0$

where $a_n$, $a_{n-1}$, ..., $a_1$, $a_0$ are constants and $n$ is a non-negative integer, which represents the degree of the polynomial. The roots of the polynomial equation are the values of $x$ that make the equation true. These roots can be real or complex numbers. In the given equation, $x^3 - 10x = -3x^2 + 24$, we have a polynomial equation of degree 3, also known as a cubic equation. Our goal is to find the three roots of this equation.

To effectively tackle the equation, we first need to rearrange it into the standard form of a polynomial equation, which means setting it equal to zero. This rearrangement will allow us to easily identify the coefficients and the degree of the polynomial, making it easier to apply different solution techniques. By bringing all the terms to one side, we create a clear and concise representation of the equation, which is crucial for both graphical and algebraic methods. This standard form not only simplifies the equation but also provides a structured framework for analyzing and solving it. The process of rearranging the equation involves adding $3x^2$ and subtracting 24 from both sides, which maintains the equality and transforms the equation into a more manageable format. This step is a fundamental aspect of solving polynomial equations, as it sets the stage for subsequent steps and ensures accurate results.

Let's solve the polynomial equation $x^3 - 10x = -3x^2 + 24$.

1. Rearrange the Equation

First, we need to rearrange the equation into the standard form of a polynomial equation, which is:

$x^3 + 3x^2 - 10x - 24 = 0$

This form allows us to easily work with the equation using both graphical and algebraic methods. By setting the equation equal to zero, we can identify the roots as the points where the graph of the polynomial intersects the x-axis. This is a critical step in solving polynomial equations because it transforms the problem into a format that is amenable to various solution techniques, including factoring, using the rational root theorem, and graphing. The standard form also helps in identifying the coefficients of the polynomial, which are essential for applying algebraic methods such as synthetic division or the quadratic formula (if the polynomial can be reduced to a quadratic). The process of rearranging the equation ensures that we have a clear and concise representation of the polynomial, which is vital for accurate analysis and solution.

2. Using a Graphing Calculator

A graphing calculator can help us visualize the equation and find the roots. We can graph the function:

$y = x^3 + 3x^2 - 10x - 24$

Steps to Graph

1. Enter the equation: Input the equation into the graphing calculator's equation editor (usually the "Y=" menu).
2. Set the window: Adjust the window settings so that the key features of the graph are visible. This often involves setting appropriate minimum and maximum values for both the x and y axes. A good starting point might be to set the x-axis from -10 to 10 and the y-axis from -50 to 50, but you may need to adjust these values based on the graph's behavior.
3. Graph the equation: Press the graph button to display the graph of the polynomial function.
4. Find the roots: Look for the points where the graph intersects the x-axis. These are the real roots of the equation. You can use the calculator's built-in functions, such as "zero" or "root," to find these points more precisely. These functions typically prompt you to set a left bound, a right bound, and a guess, which helps the calculator narrow down the search for the root within the specified interval.

Interpreting the Graph

By examining the graph, we can identify the x-intercepts, which represent the real roots of the equation. In this case, the graph will intersect the x-axis at three points, indicating that the equation has three real roots. The visual representation provided by the graphing calculator is invaluable in understanding the behavior of the polynomial function and estimating the roots. The graph not only shows the roots but also provides insights into the function's local maxima, local minima, and overall shape, which can be helpful in solving related problems or analyzing the function's properties. The process of graphing the equation and identifying the x-intercepts is a powerful tool for solving polynomial equations, especially those of higher degrees where algebraic methods can be more challenging.

3. Finding Roots from the Graph

From the graph, we can see that the roots are approximately $x = -4$, $x = -2$, and $x = 3$. These are the points where the graph intersects the x-axis, indicating that these values of $x$ make the equation $y = x^3 + 3x^2 - 10x - 24$ equal to zero. The graphical method provides a visual confirmation of the roots and helps in estimating their values. While the graph gives us approximate roots, we can use algebraic methods or calculator functions to find the exact values. The roots are the solutions to the polynomial equation, and they represent the values of $x$ that satisfy the equation. In this case, we have identified three real roots, which means there are three distinct points where the polynomial function crosses the x-axis. The graphical representation is a valuable tool in understanding the nature of the roots and the behavior of the polynomial function.

4. Verifying the Roots

To verify these roots, we can plug them back into the original equation:

1. For $x = -4$:

$(-4)^3 + 3(-4)^2 - 10(-4) - 24 = -64 + 48 + 40 - 24 = 0$

2. For $x = -2$:

$(-2)^3 + 3(-2)^2 - 10(-2) - 24 = -8 + 12 + 20 - 24 = 0$

3. For $x = 3$:

$(3)^3 + 3(3)^2 - 10(3) - 24 = 27 + 27 - 30 - 24 = 0$

Since all three values satisfy the equation, they are indeed the roots. The verification step is crucial to ensure that the values obtained from the graph or other methods are accurate solutions to the polynomial equation. By substituting the potential roots back into the equation, we can confirm whether they make the equation true. This process involves careful arithmetic and attention to detail, as any errors in the calculation can lead to incorrect conclusions. The verification step not only confirms the roots but also reinforces the understanding of what it means for a value to be a root of a polynomial equation. It demonstrates that when a root is substituted into the equation, the result is zero, which is the fundamental definition of a root. This step provides confidence in the solution and ensures that the roots are correctly identified.

5. Expressing as a System of Equations (Alternative Method)

Another approach is to express the polynomial equation as a system of equations. This method is less direct for this particular problem but illustrates a general technique.

We have:

$x^3 + 3x^2 - 10x - 24 = 0$

We can rewrite this as a system of equations, although it’s not a typical system that simplifies the solution process here.

This method is more applicable when we have two equations with two variables, and we are looking for the intersection points. In the context of a single polynomial equation, converting it into a system of equations is not the most efficient approach, but it can be a useful technique in other situations. For instance, if we had two polynomial equations, we could graph them both and find the points where they intersect, which would represent the solutions to the system. In this case, we are essentially dealing with one equation, and the best way to solve it is either by graphing the function and finding the x-intercepts or by using algebraic methods such as factoring or synthetic division. The concept of a system of equations is powerful in mathematics, but its application depends on the specific problem and the most efficient way to find the solutions.

Using a graphing calculator, we found the roots of the polynomial equation $x^3 - 10x = -3x^2 + 24$ to be $-4$, $-2$, and $3$. This corresponds to option C. The graphical method provides a visual and intuitive way to solve polynomial equations, especially those of higher degrees. While expressing the equation as a system of equations is also a valid technique, it is not the most efficient method for this particular problem. The key to solving polynomial equations is to understand the different methods available and choose the one that best suits the problem at hand. Graphing calculators are valuable tools for visualizing the solutions, but it is also important to verify the roots algebraically to ensure their accuracy. The combination of graphical and algebraic methods provides a comprehensive approach to solving polynomial equations and understanding their properties.

Therefore, the correct answer is:

C. $-4, -2, 3$