Simplifying (x^3 + 8) / (x^2 - 2x + 4) A Step-by-Step Guide

In this article, we will delve into the process of simplifying and solving the algebraic equation (x^3 + 8) / (x^2 - 2x + 4). This equation involves a rational expression, where a polynomial is divided by another polynomial. Understanding how to simplify such expressions is a fundamental skill in algebra and calculus. We will explore the techniques of factoring polynomials, specifically the sum of cubes, and polynomial long division to arrive at a simplified form and subsequently analyze potential solutions. This exploration is crucial for students and enthusiasts alike, providing a solid foundation for more advanced mathematical concepts. The ability to manipulate algebraic expressions efficiently is not just an academic exercise; it is a powerful tool in various fields, including engineering, physics, and computer science. Therefore, a thorough understanding of the steps involved in solving this equation is highly beneficial. We aim to provide a clear, step-by-step explanation that will empower you to tackle similar problems with confidence. Let's embark on this mathematical journey together, unraveling the intricacies of this equation and discovering the underlying principles that govern its behavior.

The first critical step in simplifying the expression (x^3 + 8) / (x^2 - 2x + 4) involves factoring the numerator. The numerator, x^3 + 8, is a classic example of the sum of cubes. Recognizing this pattern is key to simplifying the expression. The sum of cubes factorization follows a specific formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2). In our case, x^3 corresponds to a^3, and 8 corresponds to b^3. Therefore, we can identify a as x and b as 2 (since 2^3 = 8). Applying the sum of cubes formula, we can rewrite the numerator as (x + 2)(x^2 - 2x + 4). This factorization is a crucial step because it reveals a common factor with the denominator, which we will address in the next section. The ability to recognize and apply factoring patterns like the sum of cubes is a cornerstone of algebraic manipulation. It allows us to transform complex expressions into simpler, more manageable forms. Without this skill, simplifying rational expressions like this one would be significantly more challenging. Furthermore, understanding factoring techniques is essential for solving polynomial equations and analyzing their roots. The sum of cubes pattern is just one of several important factoring formulas, including the difference of squares and the difference of cubes, each with its own unique applications. Mastering these patterns will greatly enhance your problem-solving abilities in algebra and beyond. This initial factorization sets the stage for the subsequent simplification and eventual solution of the equation.

Now that we have factored the numerator of (x^3 + 8) / (x^2 - 2x + 4) as (x + 2)(x^2 - 2x + 4), we can proceed with simplifying the entire expression. By rewriting the original expression with the factored numerator, we have: [(x + 2)(x^2 - 2x + 4)] / (x^2 - 2x + 4). A key observation here is that the quadratic expression (x^2 - 2x + 4) appears in both the numerator and the denominator. This presents an opportunity for cancellation, a fundamental simplification technique in algebra. When a common factor exists in both the numerator and the denominator of a rational expression, it can be canceled out, provided that the factor is not equal to zero. In this case, the factor (x^2 - 2x + 4) can be canceled out, but we must acknowledge the condition that x^2 - 2x + 4 ≠ 0. This condition is crucial because division by zero is undefined in mathematics. After canceling the common factor, the expression simplifies to (x + 2). This simplified form is significantly easier to work with than the original expression. It allows us to analyze the behavior of the expression more readily and to solve for any potential values of x that satisfy a given equation. The process of simplification through cancellation highlights the power of factoring. By breaking down complex expressions into their constituent factors, we can often identify common elements that can be eliminated, leading to a more concise and manageable form. This skill is particularly valuable when dealing with rational expressions and equations, where simplification can significantly reduce the complexity of the problem.

After simplifying the original expression (x^3 + 8) / (x^2 - 2x + 4), we have arrived at the simplified form of x + 2. This linear expression is much easier to analyze than the original rational expression. The simplified form tells us that the value of the expression is simply x plus 2. This means that for any value of x, we can easily determine the value of the expression by adding 2 to it. However, it is crucial to remember the condition we established during the cancellation step: x^2 - 2x + 4 ≠ 0. This condition stems from the fact that we canceled the factor (x^2 - 2x + 4) from the numerator and the denominator. We must ensure that this factor is never equal to zero, as division by zero is undefined. To determine the values of x for which x^2 - 2x + 4 = 0, we can attempt to solve this quadratic equation. We can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by: x = [-b ± √(b^2 - 4ac)] / (2a). In our case, a = 1, b = -2, and c = 4. Plugging these values into the quadratic formula, we get: x = [2 ± √((-2)^2 - 4 * 1 * 4)] / (2 * 1). Simplifying this expression, we have: x = [2 ± √(4 - 16)] / 2 = [2 ± √(-12)] / 2. Since the discriminant (the value under the square root) is negative, the quadratic equation has no real solutions. This means that x^2 - 2x + 4 is never equal to zero for any real value of x. Therefore, the simplified form x + 2 is valid for all real numbers x. This analysis highlights the importance of considering any restrictions on the domain of the variable when simplifying expressions. Even though the simplified form is much simpler, we must remember the conditions that were imposed during the simplification process. In this case, the absence of real roots for the quadratic factor means that the simplified form is valid for all real numbers.

As we've established, simplifying the expression (x^3 + 8) / (x^2 - 2x + 4) to x + 2 involved canceling the common factor (x^2 - 2x + 4). While this simplification is mathematically sound, it's crucial to acknowledge the restrictions that this cancellation imposes on the domain of x. The fundamental principle at play here is that division by zero is undefined in mathematics. Therefore, we must ensure that the denominator of the original expression, (x^2 - 2x + 4), is never equal to zero. To determine if there are any values of x that would make the denominator zero, we need to solve the quadratic equation x^2 - 2x + 4 = 0. As we saw in the previous section, applying the quadratic formula to this equation yields complex solutions, indicating that there are no real values of x that satisfy this equation. The discriminant (b^2 - 4ac) being negative is the key indicator here, signifying that the quadratic equation has no real roots. However, it's essential to understand that the absence of real roots doesn't automatically imply that there are no restrictions. In some cases, even if a quadratic equation has no real roots, it might still have complex roots that need to be considered in certain contexts. In our specific scenario, since we are primarily dealing with real numbers, the fact that x^2 - 2x + 4 has no real roots means that it will never be equal to zero for any real value of x. Consequently, there are no real restrictions on the domain of x in this particular case. The simplified expression x + 2 is valid for all real numbers. This analysis underscores the importance of carefully examining the denominator of a rational expression before and after simplification. While cancellation simplifies the expression, it's imperative to ensure that the cancellation doesn't inadvertently introduce values of x that would make the original expression undefined. In this case, the absence of real roots for the quadratic factor in the denominator means that no such restrictions exist, and the simplified form is valid for all real numbers.

In conclusion, we have successfully simplified the algebraic expression (x^3 + 8) / (x^2 - 2x + 4) through a series of methodical steps. Our journey began with recognizing the sum of cubes pattern in the numerator, allowing us to factor it as (x + 2)(x^2 - 2x + 4). This factorization was the cornerstone of our simplification process, as it revealed a common factor with the denominator. The next crucial step involved canceling the common factor (x^2 - 2x + 4) from both the numerator and the denominator. This cancellation led to the simplified form of the expression, x + 2, which is significantly easier to analyze and manipulate. However, we also emphasized the importance of acknowledging the restrictions imposed by this cancellation. Division by zero is undefined, so we needed to ensure that the canceled factor, (x^2 - 2x + 4), is never equal to zero. To this end, we analyzed the quadratic equation x^2 - 2x + 4 = 0 and found that it has no real roots. This means that there are no real values of x that would make the denominator of the original expression equal to zero. Therefore, the simplified form x + 2 is valid for all real numbers. This comprehensive analysis demonstrates the power of algebraic manipulation techniques, such as factoring and cancellation, in simplifying complex expressions. It also highlights the importance of considering restrictions on the domain of variables when performing these operations. A thorough understanding of these concepts is essential for success in algebra and related fields. By carefully following the steps outlined in this article, you can confidently tackle similar problems and gain a deeper appreciation for the elegance and precision of mathematics. The ability to simplify and analyze algebraic expressions is a valuable skill that will serve you well in various academic and professional pursuits.