Simplifying Algebraic Expressions A Step-by-Step Guide

Introduction

In this article, we will delve into the process of simplifying the algebraic expression $\frac{-6 x^3(2 x+3)+4 x^2(-6 x-4)}{2 x^2}$. Algebraic simplification is a fundamental skill in mathematics, essential for solving equations, understanding complex functions, and tackling advanced mathematical problems. The process involves reducing an expression to its simplest form without changing its value. This often means combining like terms, factoring, and canceling common factors. Our focus will be on a step-by-step breakdown of how to simplify the given expression, making it easier to understand and manage. This process is crucial in various fields, including engineering, physics, computer science, and economics, where complex mathematical models often need simplification to yield practical solutions. A clear understanding of these techniques not only aids in academic pursuits but also enhances problem-solving skills applicable across diverse professional domains. This comprehensive guide aims to equip you with the knowledge and techniques necessary to confidently simplify algebraic expressions, enabling you to tackle more complex mathematical challenges with ease. Through detailed explanations and step-by-step solutions, we will explore the intricacies of this expression and provide a clear, concise, and effective methodology for simplification. Mastering these skills will undoubtedly pave the way for more advanced mathematical studies and applications, making this guide an invaluable resource for students and professionals alike.

Step 1: Expand the Numerator

The initial step in simplifying the expression $\frac{-6 x^3(2 x+3)+4 x^2(-6 x-4)}{2 x^2}$ involves expanding the numerator. This means applying the distributive property to remove the parentheses and combine like terms. We begin by multiplying $-6x^3$ by both terms inside the first parenthesis $(2x + 3)$. This yields $-6x^3 * 2x$ and $-6x^3 * 3$, which simplify to $-12x^4$ and $-18x^3$, respectively. Next, we multiply $4x^2$ by both terms inside the second parenthesis $(-6x - 4)$. This results in $4x^2 * -6x$ and $4x^2 * -4$, simplifying to $-24x^3$ and $-16x^2$, respectively. Now, we combine these expanded terms in the numerator: $-12x^4 - 18x^3 - 24x^3 - 16x^2$. The next task is to combine the like terms. We observe that $-18x^3$ and $-24x^3$ are like terms, as they both have the same variable and exponent. Combining these gives us $-18x^3 - 24x^3 = -42x^3$. Thus, the expanded and combined numerator is $-12x^4 - 42x^3 - 16x^2$. This step is crucial as it transforms the expression into a form where we can more easily identify common factors and simplify further. By carefully applying the distributive property and combining like terms, we set the stage for the subsequent steps in the simplification process. This meticulous approach ensures accuracy and helps in managing complexity, laying a solid foundation for the remainder of the solution.

Step 2: Factor out the Common Factor from the Numerator

After expanding and combining like terms in the numerator, our expression stands as $\frac{-12 x^4 - 42 x^3 - 16 x^2}{2 x^2}$. The next crucial step is to identify and factor out the greatest common factor (GCF) from the terms in the numerator. Factoring is the reverse process of distribution and helps to simplify expressions by isolating common elements. Looking at the coefficients -12, -42, and -16, we can identify that the greatest common numerical factor is 2. Additionally, each term in the numerator contains a power of $x$. The smallest power of $x$ present in all terms is $x^2$. Therefore, the greatest common variable factor is $x^2$. Combining these, we find that the greatest common factor for the entire numerator is $2x^2$. Now, we factor out $2x^2$ from each term in the numerator: $2x^2(-6x^2 - 21x - 8)$. Factoring out the GCF simplifies the expression and reveals a common factor between the numerator and the denominator, which is essential for further simplification. By factoring out $2x^2$, we've made the expression more manageable and set the stage for canceling common factors. This step is vital for reducing the complexity of the expression and moving towards its simplest form. The ability to identify and extract the GCF is a fundamental skill in algebra and is crucial for simplifying polynomials and rational expressions. This process not only makes the expression easier to work with but also provides insights into its structure, which can be valuable in various mathematical applications.

Step 3: Simplify the Fraction

Following the factorization of the numerator, our expression is now represented as $\frac{2 x^2(-6 x^2 - 21 x - 8)}{2 x^2}$. This step involves simplifying the fraction by canceling out common factors present in both the numerator and the denominator. This is a fundamental principle in algebra, where dividing both the numerator and the denominator by the same non-zero factor does not change the value of the expression. Upon inspection, we observe that $2x^2$ is a common factor in both the numerator and the denominator. Canceling out this common factor, we divide both the numerator and the denominator by $2x^2$. This simplifies the expression to $-6x^2 - 21x - 8$. This step is crucial for reducing the expression to its simplest form. By eliminating the common factor, we not only make the expression more concise but also easier to analyze and manipulate. The simplified expression, $-6x^2 - 21x - 8$, is a quadratic polynomial, which is much simpler to deal with compared to the original complex rational expression. This process of canceling common factors is a cornerstone of algebraic simplification and is widely used in various mathematical contexts. It highlights the importance of factoring in simplifying expressions and lays the groundwork for further analysis, such as solving equations or graphing functions. Mastering this technique is essential for students and professionals alike, as it streamlines mathematical operations and facilitates a deeper understanding of algebraic structures.

Final Answer

After meticulously expanding, factoring, and simplifying the given expression $\frac{-6 x^3(2 x+3)+4 x^2(-6 x-4)}{2 x^2}$, we have arrived at the simplified form: $-6x^2 - 21x - 8$. This final answer represents the most concise and manageable form of the original expression. Throughout the simplification process, we first expanded the numerator by applying the distributive property, which allowed us to remove parentheses and combine like terms. This step was crucial in transforming the complex expression into a more accessible format. Next, we identified and factored out the greatest common factor (GCF) from the numerator. This step not only simplified the expression but also revealed common factors between the numerator and the denominator. Finally, we canceled out the common factor present in both the numerator and the denominator, leading us to the ultimate simplified form. The resulting quadratic polynomial, $-6x^2 - 21x - 8$, is now free of any common factors and is in its simplest possible form. This process underscores the importance of each step in algebraic simplification – from expanding and combining terms to factoring and canceling. Each step plays a vital role in reducing the complexity of the expression and revealing its underlying structure. Mastering these techniques is essential for anyone working with algebraic expressions, as it allows for efficient manipulation, analysis, and problem-solving. The final simplified form not only provides a clearer understanding of the expression but also facilitates further mathematical operations, making it a crucial outcome of the simplification process.