Finding Roots: X³ - 10x = -3x² + 24 Polynomial Equation

Dec 2, 2025 by Alex Johnson 56 views

Finding the Roots of the Polynomial Equation x³ - 10x = -3x² + 24

Let's dive into the fascinating world of polynomials! In this article, we'll explore how to find the roots of the polynomial equation x³ - 10x = -3x² + 24. Polynomial equations are fundamental in mathematics, appearing in various fields like engineering, physics, and computer science. Understanding how to solve them is a crucial skill. We'll break down the steps in a clear, easy-to-follow manner, so you can confidently tackle similar problems in the future. So, grab your thinking cap, and let's get started!

Understanding Polynomial Equations

Before we jump into solving our specific equation, let's take a moment to understand what polynomial equations are and why finding their roots is so important. At its core, a polynomial equation is an equation that involves a polynomial, which is an expression consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. The general form of a polynomial equation is:

aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0

Where:

x is the variable.
n is a non-negative integer representing the degree of the polynomial.
aₙ, aₙ₋₁, ..., a₁, a₀ are the coefficients, which are constants.

In our case, the equation x³ - 10x = -3x² + 24 is a cubic polynomial equation because the highest power of x is 3. Finding the roots of a polynomial equation means finding the values of x that make the equation true, that is, the values that make the polynomial equal to zero. These roots are also known as solutions or zeros of the polynomial. Roots are incredibly significant because they reveal key information about the polynomial's behavior and graph. For example, the roots tell us where the graph of the polynomial intersects the x-axis. In practical applications, roots can represent critical points in a system, equilibrium states, or solutions to real-world problems modeled by polynomial equations. The process of finding roots can sometimes be straightforward, especially for linear and quadratic equations, but it becomes more challenging for higher-degree polynomials like cubics (degree 3) and quartics (degree 4). In such cases, various techniques, including factoring, the rational root theorem, and numerical methods, come into play. Understanding the nature and number of roots is also crucial. A polynomial of degree n has exactly n roots, counting multiplicity, in the complex number system. This is the fundamental theorem of algebra. Some roots may be real numbers, while others may be complex numbers. Real roots correspond to the x-intercepts of the polynomial's graph, while complex roots do not have a direct graphical interpretation on the real plane. In summary, understanding polynomial equations and finding their roots is a cornerstone of algebra and calculus, with far-reaching applications across various scientific and engineering disciplines. By mastering the techniques for solving these equations, you gain a powerful tool for analyzing and solving a wide range of problems.

Rewriting the Equation

The first crucial step in solving our polynomial equation, x³ - 10x = -3x² + 24, is to rearrange it into a standard form. This standard form makes it easier to apply various solution techniques, such as factoring or using the rational root theorem. The standard form of a polynomial equation is where all the terms are on one side, and the equation is set equal to zero. This allows us to clearly see the coefficients and the degree of the polynomial, which are essential for further analysis. To rewrite our equation, we need to move all the terms to the left side. We'll do this by adding 3x² to both sides and subtracting 24 from both sides. This maintains the equality of the equation while organizing it in a more manageable format. Here’s how it looks step by step:

Starting with the original equation:

x³ - 10x = -3x² + 24

Add 3x² to both sides:

x³ + 3x² - 10x = 24

Subtract 24 from both sides:

x³ + 3x² - 10x - 24 = 0

Now we have the equation in standard form: x³ + 3x² - 10x - 24 = 0. This form is much easier to work with. We can now clearly identify the coefficients: the coefficient of x³ is 1, the coefficient of x² is 3, the coefficient of x is -10, and the constant term is -24. Recognizing these coefficients is vital for applying methods like the rational root theorem, which we'll discuss later. Moreover, having the equation set to zero allows us to focus on finding the values of x that will make the left-hand side equal to zero, which are the roots of the polynomial. Rewriting the equation in standard form is a fundamental step in solving any polynomial equation. It’s like laying the foundation for a building; without it, the subsequent steps might not be as effective. This organized form provides a clear picture of the polynomial's structure, making the solution process more streamlined and less prone to errors. In the next sections, we will build upon this foundation by exploring different methods to find the roots of the equation x³ + 3x² - 10x - 24 = 0.

Applying the Rational Root Theorem

Now that we have our polynomial equation in standard form, x³ + 3x² - 10x - 24 = 0, we can use a powerful tool called the Rational Root Theorem to help us find potential roots. This theorem is particularly useful for polynomial equations with integer coefficients, like ours. The Rational Root Theorem provides a systematic way to identify possible rational roots, which are roots that can be expressed as a fraction p/q, where p and q are integers. It narrows down the possibilities, making the process of finding roots more manageable. The theorem states that if a rational number p/q is a root of the polynomial equation, then p must be a factor of the constant term (the term without any x) and q must be a factor of the leading coefficient (the coefficient of the highest power of x). In our equation, the constant term is -24, and the leading coefficient is 1. Let's identify the factors of each:

Factors of -24 (p): ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24
Factors of 1 (q): ±1

According to the Rational Root Theorem, any rational root of our equation must be of the form p/q, where p is a factor of -24 and q is a factor of 1. Since the factors of 1 are simply ±1, the possible rational roots are just the factors of -24: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24. This gives us a list of potential roots that we can test. Testing these potential roots involves substituting each value for x in the polynomial equation to see if it results in zero. If it does, then we've found a root. This process can be done by direct substitution or using synthetic division, which is often a faster method. Synthetic division is a streamlined way to divide a polynomial by a linear factor (x - r), where r is a potential root. If the remainder is zero, then r is indeed a root. For our equation, we would try each potential root one by one. For example, if we try x = 1, we substitute 1 into the equation: (1)³ + 3(1)² - 10(1) - 24 = 1 + 3 - 10 - 24 = -30, which is not zero. So, 1 is not a root. We continue this process with the other potential roots until we find one that works. The Rational Root Theorem doesn't guarantee that we'll find a root, but it significantly reduces the number of values we need to test. It's a crucial tool in our arsenal for solving polynomial equations, especially those of higher degrees. In the next section, we'll delve into the actual testing of these potential roots and see which ones satisfy our equation, bringing us closer to the complete solution.

Testing Potential Roots

Now that we've identified the potential rational roots using the Rational Root Theorem, the next step is to test these values in our polynomial equation, x³ + 3x² - 10x - 24 = 0. This is a crucial step in finding the actual roots of the equation. We'll methodically substitute each potential root for x in the equation to see if it satisfies the equation, meaning if the result is zero. There are two primary methods for testing these roots: direct substitution and synthetic division. Direct substitution involves plugging the potential root directly into the equation and evaluating. While straightforward, this method can be time-consuming, especially for higher-degree polynomials. Synthetic division, on the other hand, is a more efficient method for testing roots. It's a streamlined process for dividing a polynomial by a linear factor of the form (x - r), where r is the potential root. If the remainder after synthetic division is zero, then r is a root of the polynomial. Let's start testing our potential roots, which, as we determined earlier, are ±1, ±2, ±3, ±4, ±6, ±8, ±12, and ±24. We'll use synthetic division for efficiency.

First, let's try x = -2:

-2 | 1   3  -10  -24
    |    -2  -2   24
    ------------------
      1   1  -12   0

Since the remainder is 0, x = -2 is a root! This also gives us a new polynomial to work with: x² + x - 12. This is the quotient we obtain from the synthetic division. Now, let's try x = 3 on this new polynomial:

3 | 1   1  -12
  |    3  12
  ------------
    1   4   0

Again, the remainder is 0, so x = 3 is also a root. This leaves us with the quotient x + 4. Setting this equal to zero gives us x + 4 = 0, which means x = -4. So, we have found three roots: x = -2, x = 3, and x = -4. These are the values of x that make the original polynomial equation equal to zero. Testing potential roots is a systematic process that can sometimes require patience, especially if the roots are not integers. However, with the Rational Root Theorem and efficient methods like synthetic division, we can significantly simplify the task. By finding these roots, we've essentially solved the polynomial equation. In the next section, we will summarize our findings and discuss the implications of these roots in the context of the original problem.

The Roots of the Equation

After systematically applying the Rational Root Theorem and testing potential roots, we've successfully found the roots of the polynomial equation x³ + 3x² - 10x - 24 = 0. The roots we've identified are x = -4, x = -2, and x = 3. These are the values of x that satisfy the equation, meaning they make the equation true when substituted in. In other words, if we plug any of these values back into the original equation, the left-hand side will equal zero. Let's verify this by substituting each root back into the equation:

For x = -4:

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

For x = -2:

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

For x = 3:

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

As we can see, each root satisfies the equation, confirming our solution. The roots of a polynomial equation provide valuable information about the polynomial's behavior and graph. For a cubic equation like ours, which has a degree of 3, we expect to find up to three roots. In this case, we found exactly three real roots. Graphically, these roots represent the points where the graph of the polynomial intersects the x-axis. Knowing the roots allows us to sketch the graph of the polynomial and understand its behavior. The roots also tell us how the polynomial can be factored. Since we know the roots are -4, -2, and 3, we can write the polynomial as a product of linear factors:

(x + 4)(x + 2)(x - 3) = 0

This factored form is equivalent to the original polynomial equation. Finding the roots of a polynomial equation is not just an academic exercise; it has practical applications in various fields. Polynomial equations are used to model a wide range of phenomena in science, engineering, and economics. For example, they can describe the trajectory of a projectile, the behavior of electrical circuits, or the growth of populations. The roots of these equations often represent critical points or solutions to real-world problems. In summary, we've successfully found the roots of the polynomial equation x³ + 3x² - 10x - 24 = 0 by applying the Rational Root Theorem, testing potential roots, and verifying our solutions. These roots provide valuable insights into the behavior of the polynomial and its applications.

Conclusion

In conclusion, we've navigated the process of finding the roots of the polynomial equation x³ - 10x = -3x² + 24, which we rewrote as x³ + 3x² - 10x - 24 = 0. We started by understanding the importance of polynomial equations and their roots in mathematics and various applications. We then applied the Rational Root Theorem to identify potential rational roots, significantly narrowing down the possibilities. Next, we efficiently tested these potential roots using synthetic division, a method that streamlined the process and allowed us to quickly determine which values satisfied the equation. Through this systematic approach, we successfully identified the roots as x = -4, x = -2, and x = 3. We verified these roots by substituting them back into the original equation, confirming that they indeed make the equation true. These roots provide valuable insights into the behavior of the polynomial, including its x-intercepts and its factored form. Understanding how to find the roots of polynomial equations is a fundamental skill in mathematics, with applications ranging from engineering and physics to economics and computer science. The techniques we've explored here, such as the Rational Root Theorem and synthetic division, are powerful tools for solving polynomial equations of higher degrees. By mastering these methods, you can confidently tackle a wide range of mathematical problems and gain a deeper understanding of the world around you. Remember, practice is key to mastering these skills. The more you work with polynomial equations, the more comfortable and proficient you'll become at finding their roots. So, keep exploring, keep practicing, and keep pushing your mathematical boundaries. And for further exploration and resources on polynomial equations, consider visiting Khan Academy's Polynomial Arithmetic page for comprehensive lessons and practice exercises.