Completely Factored Form Of X²y - 2xy - 24y A Step-by-Step Solution
In the realm of algebra, factoring expressions is a fundamental skill that unlocks the door to simplifying complex equations and solving intricate problems. When confronted with the expression x²y - 2xy - 24y, the task at hand is to break it down into its constituent factors, revealing its underlying structure. This process, known as completely factoring, involves identifying and extracting all common factors, ultimately expressing the original expression as a product of simpler terms. In this comprehensive guide, we will embark on a step-by-step journey to dissect the expression x²y - 2xy - 24y, unveiling its completely factored form and shedding light on the underlying mathematical principles.
Identifying the Greatest Common Factor (GCF)
The initial step in the quest for complete factorization lies in identifying the greatest common factor (GCF) shared by all the terms within the expression. The GCF, as the name suggests, is the largest factor that divides evenly into each term. In our expression, x²y - 2xy - 24y, we observe that the variable 'y' is a common factor present in all three terms. Furthermore, we scrutinize the coefficients, the numerical values multiplying the variables. Here, we encounter the numbers 1, -2, and -24. The GCF of these coefficients is 1, as it is the largest number that divides evenly into all three. Therefore, the GCF of the entire expression is 'y'.
Having identified the GCF, we proceed to extract it from each term. This involves dividing each term by the GCF and writing the result within parentheses, preceded by the GCF. In our case, we divide each term of x²y - 2xy - 24y by 'y', yielding:
y(x² - 2x - 24)
This manipulation effectively factors out the common 'y', leaving us with a simpler quadratic expression within the parentheses.
Factoring the Quadratic Expression
Our attention now shifts to the quadratic expression x² - 2x - 24, nestled within the parentheses. Factoring a quadratic expression involves expressing it as a product of two binomials, expressions with two terms each. To achieve this, we seek two numbers that satisfy two crucial criteria:
- Their product must equal the constant term of the quadratic expression, which is -24 in our case.
- Their sum must equal the coefficient of the linear term, which is -2 in our case.
Through careful consideration, we identify the numbers 4 and -6 as the ideal candidates. Their product, 4 * (-6), equals -24, and their sum, 4 + (-6), equals -2, perfectly aligning with our criteria.
Equipped with these numbers, we can now express the quadratic expression x² - 2x - 24 as the product of two binomials:
(x + 4)(x - 6)
This factorization elegantly captures the essence of the quadratic expression, breaking it down into its fundamental building blocks.
The Completely Factored Form
Having successfully factored both the GCF and the quadratic expression, we now possess all the necessary components to construct the completely factored form of the original expression, x²y - 2xy - 24y. We simply combine the GCF, 'y', with the factored quadratic expression, (x + 4)(x - 6), to arrive at the final result:
y(x + 4)(x - 6)
This expression represents the completely factored form of x²y - 2xy - 24y, showcasing its decomposition into its simplest multiplicative components. The original expression, seemingly complex at first glance, has been elegantly transformed into a product of three factors: 'y', (x + 4), and (x - 6). This factorization not only simplifies the expression but also provides valuable insights into its mathematical behavior.
Verifying the Factored Form
To ensure the accuracy of our factorization, we can employ a simple yet effective verification technique: expansion. We expand the factored form, y(x + 4)(x - 6), using the distributive property, meticulously multiplying each term to reconstruct the original expression. Let's embark on this verification journey:
y(x + 4)(x - 6) = y(x² - 6x + 4x - 24)
Simplifying the expression within the parentheses, we get:
y(x² - 2x - 24)
Now, we distribute the 'y' across the terms within the parentheses:
y(x² - 2x - 24) = x²y - 2xy - 24y
Behold! The result of our expansion perfectly matches the original expression, x²y - 2xy - 24y, thereby confirming the accuracy of our factorization. This verification step provides a sense of assurance, solidifying our understanding of the factoring process.
Applications of Factoring
Factoring, far from being an isolated algebraic exercise, serves as a cornerstone in numerous mathematical and scientific domains. Its applications span a wide spectrum, from solving equations and simplifying expressions to modeling physical phenomena and optimizing complex systems. Let's delve into some of the key applications of factoring:
Solving Equations
Factoring plays a pivotal role in solving algebraic equations, particularly quadratic equations. By factoring a quadratic equation into the product of two binomials, we can readily identify the roots or solutions of the equation. The roots are the values of the variable that make the equation true. For instance, consider the equation:
x² - 5x + 6 = 0
Factoring the left-hand side, we get:
(x - 2)(x - 3) = 0
This equation holds true if either (x - 2) = 0 or (x - 3) = 0. Solving these simple equations, we find the roots x = 2 and x = 3. Thus, factoring transforms a seemingly intractable quadratic equation into a pair of easily solvable linear equations.
Simplifying Expressions
Factoring serves as a powerful tool for simplifying complex algebraic expressions. By identifying and extracting common factors, we can reduce the expression to its most concise form, making it easier to manipulate and understand. Consider the expression:
(x² + 3x + 2) / (x + 1)
Factoring the numerator, we get:
((x + 1)(x + 2)) / (x + 1)
Now, we can cancel the common factor (x + 1) from the numerator and denominator, resulting in the simplified expression:
x + 2
This simplification significantly reduces the complexity of the expression, making it more manageable for further analysis.
Graphing Functions
Factoring provides valuable insights into the behavior of functions, particularly polynomial functions. The factored form of a polynomial reveals its roots, the points where the graph of the function intersects the x-axis. These roots, also known as x-intercepts, are crucial in sketching the graph of the function. For example, consider the function:
f(x) = x³ - 4x
Factoring out the common factor 'x', we get:
f(x) = x(x² - 4)
Further factoring the quadratic expression, we obtain:
f(x) = x(x + 2)(x - 2)
From this factored form, we identify the roots as x = 0, x = -2, and x = 2. These roots provide us with three points where the graph of the function intersects the x-axis, aiding in the accurate sketching of the function's graph.
Real-World Applications
The applications of factoring extend far beyond the confines of theoretical mathematics, permeating various real-world scenarios. Engineers utilize factoring in structural analysis to determine the stability of bridges and buildings. Economists employ factoring in financial modeling to predict market trends and optimize investment strategies. Computer scientists leverage factoring in cryptography to secure data and protect sensitive information. The versatility of factoring underscores its significance as a fundamental tool in problem-solving across diverse disciplines.
Conclusion
In this comprehensive exploration, we have meticulously dissected the expression x²y - 2xy - 24y, unveiling its completely factored form as y(x + 4)(x - 6). Our journey began with identifying the greatest common factor, 'y', followed by factoring the quadratic expression x² - 2x - 24 into the product of two binomials, (x + 4)(x - 6). We then combined these factors to arrive at the completely factored form, which we subsequently verified through expansion.
Furthermore, we delved into the diverse applications of factoring, highlighting its significance in solving equations, simplifying expressions, graphing functions, and modeling real-world phenomena. Factoring, far from being an abstract mathematical concept, serves as a powerful tool for simplifying complexity, revealing hidden structures, and unlocking solutions across a wide range of disciplines. By mastering the art of factoring, we equip ourselves with a fundamental skill that empowers us to tackle intricate problems and navigate the intricate world of mathematics and its applications.
Is y(x+4)(x-6) the completely factored form of x²y-2xy-24y?
Yes, y(x+4)(x-6) is the completely factored form of the expression x²y - 2xy - 24y. We arrived at this solution through a step-by-step process, ensuring each factor is broken down to its simplest form. This complete factorization allows for easier simplification, solving equations, and understanding the behavior of related functions.
