Factoring $216 Y^3-348 X^3$ Using Difference Of Cubes Pattern

Jul 22, 2025 by ADMIN 62 views

Factoring the Difference of Cubes: A Comprehensive Guide to $216y^3 - 348x^3$

Introduction to Factoring and the Difference of Cubes

In the realm of mathematics, factoring is a fundamental skill that allows us to break down complex expressions into simpler, more manageable components. This process is particularly useful in solving equations, simplifying expressions, and gaining a deeper understanding of the underlying structure of mathematical relationships. One of the most intriguing and frequently encountered factoring patterns is the difference of cubes. This pattern arises when we have two perfect cubes that are being subtracted from each other, and it offers a structured approach to factoring such expressions completely.

When we talk about factoring the difference of cubes, we are essentially looking for two expressions, a binomial and a trinomial, whose product equals the original expression. The binomial part represents the difference of the cube roots of the two terms, while the trinomial part is derived from the binomial using a specific pattern. Mastering this pattern not only simplifies factoring but also enhances our algebraic manipulation skills, which are crucial in various mathematical contexts. In this comprehensive guide, we will delve into the intricacies of factoring the difference of cubes, using the expression $216y^3 - 348x^3$ as our primary example. We will explore each step in detail, from identifying the pattern to applying the formula and simplifying the result. By the end of this guide, you will have a solid understanding of how to factor the difference of cubes and be able to apply this knowledge to a wide range of similar problems. This is a critical skill for anyone studying algebra, precalculus, or calculus, as it forms the basis for many advanced mathematical techniques. So, let's embark on this mathematical journey and unravel the secrets of factoring the difference of cubes!

Identifying the Difference of Cubes Pattern

Before we can apply the difference of cubes formula, we must first identify whether the given expression fits the pattern. The difference of cubes pattern is characterized by two key features: first, there must be two terms, each of which is a perfect cube; second, these terms must be separated by a subtraction sign. A perfect cube is a number or variable that can be expressed as the cube of another number or variable. For example, 8 is a perfect cube because it can be written as $2^3$ (2 cubed), and $x^3$ is a perfect cube because it is the cube of $x$ . In the expression $216y^3 - 348x^3$ , we need to determine if both $216y^3$ and $348x^3$ are perfect cubes. Let's break down each term individually. For $216y^3$ , we need to check if 216 is a perfect cube and if $y^3$ is a perfect cube. The number 216 is indeed a perfect cube, as $6^3 = 6 imes 6 imes 6 = 216$ . The variable term $y^3$ is also a perfect cube, as it is simply $y$ raised to the power of 3. So, $216y^3$ can be expressed as $(6y)^3$ . Now, let's consider $348x^3$ . The variable term $x^3$ is clearly a perfect cube, but we need to determine if 348 is a perfect cube. To do this, we can try to find an integer whose cube is 348. We know that $7^3 = 343$ , which is close to 348, but not equal. Since $8^3 = 512$ , which is greater than 348, we can conclude that 348 is not a perfect cube. This means that the expression $216y^3 - 348x^3$ does not initially fit the difference of cubes pattern in its current form. However, we can simplify it further by factoring out the greatest common factor (GCF) from both terms, which will help us reveal the perfect cubes. Factoring out the GCF is a crucial step in many factoring problems, as it often simplifies the expression and makes it easier to apply other factoring techniques. In the next section, we will explore how to factor out the GCF from $216y^3 - 348x^3$ and see if we can then apply the difference of cubes pattern. Identifying the difference of cubes pattern correctly is the first step towards successful factoring, and it requires careful attention to the structure of the expression and the properties of perfect cubes.

Factoring Out the Greatest Common Factor (GCF)

As we determined in the previous section, the expression $216y^3 - 348x^3$ does not immediately present itself as a straightforward difference of cubes due to the coefficient 348 not being a perfect cube. However, a crucial step in factoring any expression is to first identify and factor out the greatest common factor (GCF). The GCF is the largest factor that divides evenly into all terms of the expression. In this case, we need to find the GCF of 216 and 348. One way to find the GCF is to list the factors of each number and identify the largest factor they have in common. The factors of 216 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, and 216. The factors of 348 are 1, 2, 3, 4, 6, 12, 29, 58, 87, 116, 174, and 348. By comparing these lists, we can see that the greatest common factor of 216 and 348 is 12. Now that we have identified the GCF as 12, we can factor it out from the expression $216y^3 - 348x^3$ . This involves dividing each term by 12 and writing the expression as a product of the GCF and the resulting expression in parentheses. So, we have: $216y^3 - 348x^3 = 12( rac{216y^3}{12} - rac{348x^3}{12})$ . Performing the divisions, we get: $216y^3 / 12 = 18y^3$ and $348x^3 / 12 = 29x^3$ . Therefore, the expression becomes: $12(18y^3 - 29x^3)$ . Now, we need to re-examine the expression inside the parentheses, $18y^3 - 29x^3$ , to see if it fits the difference of cubes pattern. We need to check if 18 and 29 are perfect cubes. As we discussed earlier, a perfect cube is a number that can be obtained by cubing an integer. Since 18 and 29 are not perfect cubes, the expression $18y^3 - 29x^3$ cannot be factored further using the difference of cubes pattern directly. However, factoring out the GCF has simplified the expression and made it more manageable. Factoring out the GCF is a critical first step in many factoring problems, as it often reveals underlying patterns or simplifies the expression to a form that is easier to work with. In this case, while it didn't directly lead to the difference of cubes factorization, it did reduce the coefficients and make the expression simpler. This is a valuable skill in algebra, as it allows us to break down complex expressions into more manageable parts. In the next section, we will discuss the general formula for factoring the difference of cubes and how it applies to expressions that fit the pattern perfectly.

Applying the Difference of Cubes Formula

Now that we have factored out the greatest common factor from the original expression, we have $12(18y^3 - 29x^3)$ . Unfortunately, the expression inside the parentheses, $18y^3 - 29x^3$ , does not fit the difference of cubes pattern because 18 and 29 are not perfect cubes. However, it's essential to understand the difference of cubes formula for cases where it does apply. The difference of cubes formula is a powerful tool for factoring expressions of the form $A^3 - B^3$ , where A and B can be any algebraic terms. The formula states that: $A^3 - B^3 = (A - B)(A^2 + AB + B^2)$ . This formula provides a direct way to factor the difference of two perfect cubes into a binomial (A - B) and a trinomial ( $A^2 + AB + B^2$ ). To apply this formula, we first need to identify A and B, which are the cube roots of the two terms in the expression. For example, if we had an expression like $8x^3 - 27y^3$ , we would recognize that $8x^3$ is $(2x)^3$ and $27y^3$ is $(3y)^3$ . Thus, A would be $2x$ and B would be $3y$ . Once we have identified A and B, we can substitute them into the formula: $(2x)^3 - (3y)^3 = (2x - 3y)((2x)^2 + (2x)(3y) + (3y)^2)$ . Then, we simplify the terms in the trinomial: $(2x - 3y)(4x^2 + 6xy + 9y^2)$ . This gives us the completely factored form of the expression. Let's break down the formula further to understand why it works. The binomial part, $(A - B)$ , represents the difference of the cube roots of the two terms. The trinomial part, $(A^2 + AB + B^2)$ , is derived from squaring the first term (A), adding the product of the two terms (A and B), and adding the square of the second term (B). This trinomial is carefully constructed so that when it is multiplied by the binomial (A - B), the middle terms cancel out, leaving only the difference of the cubes. The difference of cubes formula is a valuable tool in algebra because it allows us to factor expressions that would be difficult or impossible to factor using other methods. It is particularly useful in solving cubic equations, simplifying algebraic fractions, and working with more advanced mathematical concepts. While the expression $18y^3 - 29x^3$ cannot be factored using this formula directly, understanding the formula is crucial for recognizing and factoring other expressions that do fit the pattern. In the next section, we will recap the steps we have taken so far and discuss some additional strategies for factoring algebraic expressions.

Conclusion and Summary of Steps

In this comprehensive guide, we set out to factor the expression $216y^3 - 348x^3$ completely. While the initial expression did not immediately fit the difference of cubes pattern, we embarked on a systematic approach to factoring, which is a valuable skill in algebra and beyond. Here's a summary of the steps we took: 1. Identifying the Difference of Cubes Pattern: We first examined the expression to see if it fit the form $A^3 - B^3$ . We determined that while the variable terms $y^3$ and $x^3$ were perfect cubes, the coefficients 216 and 348 needed further examination. 2. Factoring Out the Greatest Common Factor (GCF): Recognizing that 348 is not a perfect cube, we identified and factored out the greatest common factor (GCF) of 216 and 348, which is 12. This simplified the expression to $12(18y^3 - 29x^3)$ . 3. Applying the Difference of Cubes Formula: We then considered the expression inside the parentheses, $18y^3 - 29x^3$ , and determined that it does not fit the difference of cubes pattern because 18 and 29 are not perfect cubes. We discussed the difference of cubes formula, $A^3 - B^3 = (A - B)(A^2 + AB + B^2)$ , and how it applies to expressions that do fit the pattern. 4. Final Factored Form: Since $18y^3 - 29x^3$ cannot be factored further using elementary methods, the completely factored form of the original expression is $12(18y^3 - 29x^3)$ . In conclusion, while we couldn't apply the difference of cubes formula directly to the original expression, we learned the importance of factoring out the GCF as a first step. This often simplifies the expression and may reveal underlying patterns. We also reinforced our understanding of the difference of cubes formula and how to apply it when the pattern is present. Factoring is a fundamental skill in mathematics, and mastering different factoring techniques is crucial for solving a wide range of problems. By following a systematic approach and understanding the various factoring patterns, you can confidently tackle complex expressions and simplify them into more manageable forms. This guide has provided a detailed walkthrough of the process, and with practice, you can become proficient in factoring algebraic expressions. Always remember to look for the GCF first, then consider other factoring patterns such as the difference of cubes, sum of cubes, and difference of squares. With these tools in your arsenal, you'll be well-equipped to handle any factoring challenge that comes your way.