What Is The Completely Factored Form Of 2x² - 32? Show Step By Step.
Let's embark on a journey into the realm of algebra, where we'll unravel the mystery behind factoring the quadratic expression 2x² - 32. This seemingly simple expression holds a secret, a hidden structure that we can reveal through the art of factorization. In this comprehensive guide, we'll not only dissect the process step-by-step but also explore the underlying principles that make factorization such a powerful tool in mathematics. Factoring is not just a mechanical procedure; it's a way of understanding the composition of mathematical expressions, allowing us to simplify complex problems and gain deeper insights into their behavior. This particular expression, 2x² - 32, presents a classic example of a difference of squares, a pattern that frequently appears in algebraic manipulations. By recognizing and applying the appropriate factoring techniques, we can transform this expression into its most fundamental form, revealing its true nature.
The Quest for Complete Factorization
Before we dive into the specifics of 2x² - 32, let's establish a clear understanding of what complete factorization entails. In essence, complete factorization means breaking down an expression into its prime factors, much like decomposing a number into its prime components. For polynomials, this involves expressing them as a product of simpler polynomials or constants that cannot be factored further. This process is crucial for solving equations, simplifying expressions, and gaining a deeper understanding of mathematical relationships. When we say an expression is completely factored, we mean that it has been reduced to its most basic building blocks, leaving no room for further decomposition. This ensures that we have captured the essence of the expression in its most concise and revealing form. The quest for complete factorization is not merely an exercise in algebraic manipulation; it's a journey towards mathematical clarity and understanding. By mastering these techniques, we unlock the power to simplify complex problems and gain deeper insights into the world of mathematics.
Spotting the Common Factor: The First Step
The first step in factoring any expression is to identify and extract the greatest common factor (GCF) of its terms. In our case, the expression is 2x² - 32. Looking at the coefficients, we see that both 2 and 32 are divisible by 2. Furthermore, the first term contains x², while the second term is a constant. Therefore, the GCF is simply 2. Factoring out the 2, we rewrite the expression as 2(x² - 16). This seemingly small step is crucial, as it simplifies the expression and sets the stage for further factorization. By extracting the common factor, we reduce the complexity of the problem and reveal the underlying structure more clearly. This initial step is a cornerstone of factoring, and mastering it will significantly enhance your ability to tackle more complex expressions. Recognizing and extracting the GCF is not just a mechanical procedure; it's an exercise in pattern recognition and mathematical intuition. By developing this skill, you'll be able to quickly identify the common threads that run through algebraic expressions, making the process of factorization more efficient and intuitive.
Unveiling the Difference of Squares Pattern
Now, let's turn our attention to the expression inside the parentheses: (x² - 16). This expression holds a special significance because it fits a well-known pattern called the difference of squares. The difference of squares pattern states that for any two terms, a² and b², their difference can be factored as a² - b² = (a + b)(a - b). This pattern is a cornerstone of algebraic manipulation, and recognizing it can greatly simplify the factoring process. In our case, x² corresponds to a², and 16 corresponds to b² (since 16 = 4²). Therefore, we can apply the difference of squares pattern to factor (x² - 16). By recognizing this pattern, we transform a seemingly complex expression into a simple product of two binomials. This highlights the power of pattern recognition in mathematics, allowing us to apply known formulas and techniques to solve problems efficiently. The difference of squares pattern is not just a mathematical trick; it's a reflection of a fundamental algebraic relationship. By understanding this relationship, we can not only factor expressions but also gain a deeper appreciation for the interconnectedness of mathematical concepts.
Applying the Difference of Squares Formula
Applying the difference of squares formula, we substitute a = x and b = 4 into the equation a² - b² = (a + b)(a - b). This gives us x² - 16 = (x + 4)(x - 4). Now we have successfully factored the expression inside the parentheses. This step demonstrates the power of applying known formulas to simplify mathematical expressions. By recognizing the difference of squares pattern and applying the corresponding formula, we efficiently transformed a seemingly complex expression into its factored form. This highlights the importance of memorizing key formulas and understanding their applications in various mathematical contexts. Applying the difference of squares formula is not just a mechanical substitution; it's an exercise in algebraic manipulation and pattern recognition. By mastering this technique, you'll be able to quickly factor expressions that fit this pattern, saving time and effort in problem-solving. The ability to recognize and apply such formulas is a hallmark of mathematical proficiency, allowing you to navigate complex problems with confidence and efficiency.
The Grand Finale: Complete Factorization Achieved
Finally, we combine the GCF we factored out earlier with the factored form of (x² - 16). Recall that we had 2(x² - 16), and we just factored (x² - 16) into (x + 4)(x - 4). Therefore, the completely factored form of 2x² - 32 is 2(x + 4)(x - 4). This is the final answer, representing the expression in its most simplified and factored form. We have successfully broken down the original expression into its prime factors, revealing its underlying structure and composition. This process demonstrates the power of factorization as a tool for simplifying expressions and gaining mathematical insights. The journey to complete factorization is not just about finding the right answer; it's about developing a systematic approach to problem-solving and a deeper understanding of mathematical concepts. By mastering these techniques, you'll be able to tackle a wide range of algebraic problems with confidence and efficiency. The completely factored form, 2(x + 4)(x - 4), represents the culmination of our efforts, showcasing the elegance and power of mathematical manipulation.
Verifying the Result: A Crucial Step
To ensure the accuracy of our factorization, it's always a good practice to verify the result. We can do this by expanding the factored expression and checking if it matches the original expression. Expanding 2(x + 4)(x - 4), we first multiply (x + 4) and (x - 4), which gives us (x² - 16). Then, we multiply the result by 2, obtaining 2(x² - 16) = 2x² - 32. This matches our original expression, confirming that our factorization is correct. Verification is a crucial step in the mathematical process, ensuring that we haven't made any errors along the way. It's a safeguard against mistakes and a way to build confidence in our solutions. By expanding the factored expression, we effectively reverse the factorization process, allowing us to check our work and validate our answer. This process not only ensures accuracy but also reinforces our understanding of the relationship between factored and expanded forms. The practice of verification is a hallmark of a meticulous mathematician, demonstrating a commitment to precision and a desire for a thorough understanding of the problem.
The Answer and Its Significance
Therefore, the completely factored form of 2x² - 32 is 2(x + 4)(x - 4), which corresponds to option B. This seemingly simple result holds significant implications in various mathematical contexts. Factoring expressions allows us to solve equations, simplify complex formulas, and gain deeper insights into the behavior of mathematical functions. For example, if we were to solve the equation 2x² - 32 = 0, the factored form 2(x + 4)(x - 4) = 0 immediately reveals the solutions: x = -4 and x = 4. This demonstrates the power of factorization as a tool for solving equations and finding roots. Furthermore, the factored form can provide valuable information about the graph of a function represented by the expression. The roots of the equation correspond to the x-intercepts of the graph, providing key points for sketching the curve. The significance of complete factorization extends beyond mere algebraic manipulation; it's a fundamental concept that underpins many areas of mathematics and its applications. By mastering factorization techniques, we unlock the power to solve problems, simplify expressions, and gain a deeper appreciation for the interconnectedness of mathematical ideas. The factored form, 2(x + 4)(x - 4), is not just an answer; it's a gateway to a deeper understanding of the mathematical world.
Mastering Factorization: A Key to Mathematical Success
In conclusion, we've successfully navigated the process of factoring 2x² - 32, arriving at the completely factored form 2(x + 4)(x - 4). This journey has not only provided us with a specific answer but has also illuminated the broader principles of factorization. Factoring is not merely a mechanical procedure; it's a fundamental skill that empowers us to simplify expressions, solve equations, and gain deeper insights into mathematical relationships. By mastering factorization techniques, we unlock the potential to tackle a wide range of mathematical challenges with confidence and efficiency. The ability to recognize patterns, extract common factors, and apply formulas like the difference of squares is a hallmark of mathematical proficiency. Furthermore, the practice of verifying our results ensures accuracy and reinforces our understanding of the underlying concepts. Mastering factorization is not just about getting the right answer; it's about developing a systematic approach to problem-solving and a deeper appreciation for the elegance and power of mathematics. The journey from the original expression, 2x² - 32, to its completely factored form, 2(x + 4)(x - 4), is a testament to the transformative power of algebraic manipulation and the importance of mastering fundamental mathematical skills.