Solve The Equation X³ + 5x² - 3x - 15 = 0 For X.

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Solving cubic equations can seem daunting, but there are several methods we can employ to find the roots. In this article, we'll explore one such method: factoring by grouping. This technique allows us to break down the cubic equation into simpler quadratic expressions, ultimately leading us to the solutions. Let's dive into the step-by-step process of solving the given equation:

1. Understanding the Cubic Equation

The equation we aim to solve is a cubic equation, specifically:

x³ + 5x² - 3x - 15 = 0

A cubic equation is a polynomial equation where the highest power of the variable (in this case, x) is 3. Cubic equations can have up to three solutions, which can be real or complex numbers. Our goal is to find all the values of x that satisfy this equation. Factoring by grouping is a powerful technique that simplifies this process, by helping us identify common factors and reduce the complexity of the equation. The effectiveness of this method hinges on our ability to strategically group terms and extract shared factors, thus paving the way for a more manageable equation that can be solved using standard algebraic methods.

2. Factoring by Grouping

The key to solving this equation lies in the method of factoring by grouping. This technique involves rearranging the terms and factoring out common factors from pairs of terms. Let's break down the steps:

Step 1: Grouping Terms

First, we group the terms in pairs:

(x³ + 5x²) + (-3x - 15) = 0

Grouping the terms allows us to visually organize the equation and identify potential common factors within each group. This is a critical first step in simplifying the equation, as it sets the stage for extracting these factors and reducing the complexity of the expression. The choice of grouping can sometimes be guided by the coefficients of the terms, looking for combinations that might lead to shared factors. However, in this case, the natural grouping of the first two terms and the last two terms works perfectly for our factoring strategy.

Step 2: Factoring out Common Factors

Next, we factor out the greatest common factor (GCF) from each group:

From the first group (x³ + 5x²), the GCF is x². Factoring this out, we get: x²(x + 5)

From the second group (-3x - 15), the GCF is -3. Factoring this out, we get: -3(x + 5)

Now our equation looks like this:

x²(x + 5) - 3(x + 5) = 0

Factoring out the greatest common factor from each group is the heart of this method. By identifying and extracting the GCF, we are essentially reversing the distributive property, which allows us to rewrite each group as a product of factors. This step is crucial because it reveals a common binomial factor across the two groups, which is the key to further simplification. The ability to spot these common factors comes with practice, but it is a skill that greatly enhances problem-solving in algebra.

Step 3: Factoring out the Common Binomial

Notice that both terms now have a common binomial factor of (x + 5). We can factor this out:

(x + 5)(x² - 3) = 0

Factoring out the common binomial is the pivotal step that transforms the cubic equation into a product of simpler factors. This simplification is significant because it allows us to apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. By recognizing and extracting the common binomial, we effectively reduce the problem to solving two lower-degree equations, making the solutions much more accessible.

3. Applying the Zero-Product Property

The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This is a fundamental principle in algebra that allows us to solve equations that are factored. In our case, we have the equation:

(x + 5)(x² - 3) = 0

This means either (x + 5) = 0 or (x² - 3) = 0, or both. Now, let's solve each equation separately.

Case 1: x + 5 = 0

Subtracting 5 from both sides, we get:

x = -5

This gives us our first solution. It's a straightforward linear equation, and the solution is easily obtained by isolating x. This solution represents one of the roots of the original cubic equation.

Case 2: x² - 3 = 0

Adding 3 to both sides, we get:

x² = 3

Taking the square root of both sides, we get:

x = ±√3

This gives us two more solutions: x = √3 and x = -√3. Remember that when taking the square root of both sides of an equation, we must consider both the positive and negative roots. These two solutions, along with the solution from Case 1, represent all the roots of the original cubic equation.

4. The Solutions

Therefore, the values of x that satisfy the equation x³ + 5x² - 3x - 15 = 0 are:

  • x = -5
  • x = √3
  • x = -√3

These are the three roots of the cubic equation. We have successfully found them by using the factoring by grouping method and the zero-product property. These solutions are the points where the graph of the cubic function intersects the x-axis. They are crucial for understanding the behavior of the function and its applications in various fields of mathematics and science. The process of finding these roots demonstrates the power of algebraic techniques in solving complex equations.

5. Verification (Optional but Recommended)

To ensure our solutions are correct, we can substitute each value of x back into the original equation and check if it holds true. This is a good practice to avoid errors and build confidence in our results.

Verification for x = -5:

(-5)³ + 5(-5)² - 3(-5) - 15 = -125 + 125 + 15 - 15 = 0 (Correct)

Verification for x = √3:

(√3)³ + 5(√3)² - 3(√3) - 15 = 3√3 + 15 - 3√3 - 15 = 0 (Correct)

Verification for x = -√3:

(-√3)³ + 5(-√3)² - 3(-√3) - 15 = -3√3 + 15 + 3√3 - 15 = 0 (Correct)

Since all three values satisfy the original equation, we can confidently say that our solutions are correct. This verification step is crucial in mathematical problem-solving, as it provides a means to confirm the accuracy of our results and detect any potential errors in our calculations. It reinforces the understanding that the solutions we have found are indeed the roots of the given equation.

6. Conclusion

In summary, we have successfully solved the cubic equation x³ + 5x² - 3x - 15 = 0 by employing the technique of factoring by grouping. This method allowed us to rewrite the equation as a product of simpler factors, which we then solved using the zero-product property. We found three solutions:

  • x = -5
  • x = √3
  • x = -√3

These values represent the roots of the cubic equation. Factoring by grouping is a valuable algebraic technique, especially useful for polynomials with four terms where common factors can be identified. This method not only helps in finding solutions but also enhances our understanding of polynomial structures and factorization principles. Mastering such techniques is essential for tackling more complex algebraic problems and is a fundamental skill in mathematics. The process we followed illustrates a systematic approach to solving cubic equations, which can be applied to similar problems in the future.