Simplifying (x+y)^3 - 64 A Step-by-Step Guide

In the realm of algebra, simplifying expressions is a fundamental skill. It allows us to rewrite complex equations into a more manageable and understandable form. This article delves into the process of simplifying the algebraic expression $(x+y)^3 - 64$, providing a step-by-step guide that caters to both beginners and those seeking a refresher. Our main goal is to transform the given expression into its simplest factored form, which not only reveals its structure but also makes it easier to solve related equations or analyze its behavior.

The expression $(x+y)^3 - 64$ initially presents itself as a difference of cubes. Recognizing this pattern is the key to unlocking its simplification. The difference of cubes identity, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, is our primary tool in this endeavor. By carefully identifying 'a' and 'b' within our expression, we can apply this identity to break down the complex expression into simpler factors. This process not only simplifies the expression but also provides a deeper understanding of its underlying algebraic structure. Let's embark on this simplification journey together, unraveling the intricacies of this expression and mastering the art of algebraic simplification. We will explore each step with clarity, ensuring that the logic and reasoning behind every transformation are crystal clear.

Recognizing the Difference of Cubes Pattern

The cornerstone of simplifying $(x+y)^3 - 64$ lies in recognizing it as a difference of cubes. This algebraic pattern is crucial for factoring expressions of the form $a^3 - b^3$. To effectively apply this pattern, we must first identify 'a' and 'b' within our given expression. In our case, $(x+y)^3$ directly corresponds to $a^3$, making $(x+y)$ the 'a' term. The constant term, 64, can be expressed as $4^3$, thus making 4 the 'b' term. This identification is the crucial first step, as it allows us to directly apply the difference of cubes factorization formula.

The difference of cubes identity states that $a^3 - b^3$ can be factored into $(a - b)(a^2 + ab + b^2)$. Understanding this identity is paramount, as it provides the framework for our simplification process. It essentially breaks down a complex cubic expression into a product of a linear term $(a - b)$ and a quadratic term $(a^2 + ab + b^2)$. Recognizing the structure of this identity and how it applies to our specific problem is key to successfully simplifying the expression. The ability to identify such patterns and apply relevant algebraic identities is a fundamental skill in mathematics, enabling us to tackle a wide range of simplification and factorization problems.

Applying the Difference of Cubes Formula

Now that we've identified the difference of cubes pattern in $(x+y)^3 - 64$ and recognized that $a = (x+y)$ and $b = 4$, we can confidently apply the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Substituting our 'a' and 'b' values into the formula, we get:

$(x+y)^3 - 64 = [(x+y) - 4][(x+y)^2 + (x+y)(4) + 4^2]$

This substitution is a crucial step, transforming our original expression into a product of two factors. The first factor, $(x+y) - 4$, is a straightforward linear expression. The second factor, $(x+y)^2 + (x+y)(4) + 4^2$, is a quadratic expression that requires further simplification. This step exemplifies the power of algebraic identities in breaking down complex expressions into more manageable components. By applying the difference of cubes formula, we've effectively taken a cubic expression and expressed it as a product of a linear and a quadratic factor, paving the way for further simplification and analysis. This application not only simplifies the expression but also provides valuable insights into its structure and potential solutions.

Expanding and Simplifying the Quadratic Factor

Having applied the difference of cubes formula, we've arrived at the expression: $[(x+y) - 4][(x+y)^2 + (x+y)(4) + 4^2]$. The next step involves expanding and simplifying the quadratic factor, $(x+y)^2 + (x+y)(4) + 4^2$. To do this, we'll first expand $(x+y)^2$ using the binomial expansion formula, $(a+b)^2 = a^2 + 2ab + b^2$. This gives us $x^2 + 2xy + y^2$.

Next, we'll distribute the 4 in the term $(x+y)(4)$, resulting in $4x + 4y$. Finally, we calculate $4^2$, which is 16. Substituting these expansions back into the quadratic factor, we get:

$x^2 + 2xy + y^2 + 4x + 4y + 16$

This expanded form of the quadratic factor now needs to be simplified by combining any like terms. In this case, there are no like terms to combine, so the expression remains as $x^2 + 2xy + y^2 + 4x + 4y + 16$. This step is crucial in achieving the most simplified form of the original expression. By carefully expanding and simplifying the quadratic factor, we ensure that our final expression is as concise and understandable as possible. This meticulous approach to simplification is a hallmark of strong algebraic skills.

The Fully Simplified Expression

After expanding and simplifying the quadratic factor, we can now present the fully simplified form of the original expression, $(x+y)^3 - 64$. We started by applying the difference of cubes formula, which gave us $[(x+y) - 4][(x+y)^2 + (x+y)(4) + 4^2]$. We then expanded and simplified the quadratic factor, $(x+y)^2 + (x+y)(4) + 4^2$, to obtain $x^2 + 2xy + y^2 + 4x + 4y + 16$. Combining this simplified quadratic factor with the linear factor, $(x+y) - 4$, we arrive at the final simplified expression:

$[(x+y) - 4][x^2 + 2xy + y^2 + 4x + 4y + 16]$

This factored form is the most simplified representation of the original expression. It clearly shows the factors that make up the expression, providing valuable insights into its algebraic structure. This final step demonstrates the power of systematic simplification, where each step builds upon the previous one to achieve a concise and understandable result. The ability to simplify expressions to their fullest extent is a fundamental skill in algebra, enabling us to solve equations, analyze functions, and tackle more complex mathematical problems.

In this comprehensive guide, we've successfully simplified the algebraic expression $(x+y)^3 - 64$. We began by recognizing the difference of cubes pattern, a crucial step in unlocking the simplification process. We then applied the difference of cubes formula, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, to factor the expression into $[(x+y) - 4][(x+y)^2 + (x+y)(4) + 4^2]$. The next step involved expanding and simplifying the quadratic factor, $(x+y)^2 + (x+y)(4) + 4^2$, resulting in $x^2 + 2xy + y^2 + 4x + 4y + 16$. Finally, we combined the simplified factors to arrive at the fully simplified expression:

$[(x+y) - 4][x^2 + 2xy + y^2 + 4x + 4y + 16]$

This journey through algebraic simplification highlights the importance of recognizing patterns, applying appropriate formulas, and meticulously expanding and simplifying expressions. Mastering these skills is essential for success in algebra and beyond. By understanding the underlying principles and practicing regularly, you can confidently tackle a wide range of algebraic simplification problems. This process not only enhances your mathematical abilities but also fosters critical thinking and problem-solving skills that are valuable in various aspects of life. Algebraic simplification is more than just manipulating symbols; it's about understanding the structure and relationships within mathematical expressions, empowering you to solve complex problems with clarity and precision.