Factoring X² - 100 A Step-by-Step Guide

The realm of algebra often presents us with expressions that seem daunting at first glance. However, with a systematic approach and a solid understanding of fundamental concepts, we can unravel even the most intricate mathematical puzzles. In this article, we will delve into the process of factoring the quadratic expression x² - 100, exploring the underlying principles and techniques that lead us to the correct solution. This problem serves as an excellent example to illustrate the power of algebraic manipulation and the importance of recognizing patterns in mathematical expressions. By the end of this exploration, you will not only be able to factor this specific expression but also gain a deeper appreciation for the art of algebraic factorization.

Understanding the Basics of Factoring

At its core, factoring is the process of breaking down a mathematical expression into its constituent parts, typically smaller expressions that, when multiplied together, yield the original expression. In the context of quadratic expressions, which take the general form ax² + bx + c, factoring involves finding two binomials (expressions with two terms) that multiply to give the quadratic. Factoring quadratic expressions is a fundamental skill in algebra, serving as a building block for solving equations, simplifying expressions, and tackling more advanced mathematical concepts. There are several techniques for factoring, including the greatest common factor (GCF) method, the difference of squares pattern, and the quadratic formula. The choice of method often depends on the specific characteristics of the expression being factored.

Recognizing the Difference of Squares Pattern

The expression x² - 100 presents a unique opportunity to apply a specific factoring pattern known as the difference of squares. This pattern arises when we have an expression in the form a² - b², where 'a' and 'b' are any algebraic terms. The beauty of the difference of squares pattern lies in its straightforward factorization: a² - b² = (a - b)(a + b). Recognizing this pattern can significantly simplify the factoring process, allowing us to bypass more complex methods. In our case, x² - 100 perfectly aligns with this pattern. We can identify x² as a² and 100 as b², where a = x and b = 10. This recognition is the key to unlocking the factorization of the expression.

Applying the Difference of Squares to x² - 100

Now that we have identified the difference of squares pattern, applying it to x² - 100 becomes a simple task. We know that a = x and b = 10, so we can directly substitute these values into the formula a² - b² = (a - b)(a + b). This gives us x² - 100 = (x - 10)(x + 10). This elegant solution demonstrates the power of pattern recognition in mathematics. By recognizing the difference of squares, we have effortlessly factored the quadratic expression into two binomials. Each binomial represents a factor of the original expression, and when multiplied together, they will result in x² - 100. This factorization is not only mathematically sound but also provides valuable insights into the structure and properties of the expression.

Verifying the Factors: Expanding (x - 10)(x + 10)

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 (x - 10)(x + 10) and checking if it matches the original expression x² - 100. Expanding the expression involves multiplying each term in the first binomial by each term in the second binomial. This can be done using the distributive property or the FOIL (First, Outer, Inner, Last) method. Let's apply the FOIL method:

• First: x * x = x²
• Outer: x * 10 = 10x
• Inner: -10 * x = -10x
• Last: -10 * 10 = -100

Adding these terms together, we get x² + 10x - 10x - 100. Notice that the 10x and -10x terms cancel each other out, leaving us with x² - 100. This confirms that our factorization (x - 10)(x + 10) is indeed correct. The verification process not only validates our solution but also reinforces our understanding of the relationship between factors and the original expression.

Why Other Options Are Incorrect

Let's examine why the other options provided are not correct factors of x² - 100:

• A. (x - 50)(x + 50): Expanding this expression gives us x² - 2500, which is not equal to x² - 100. This option fails to correctly apply the difference of squares pattern.
• C. (x - 25)(x + 4): Expanding this expression gives us x² - 21x - 100, which is also not equal to x² - 100. This option demonstrates an incorrect understanding of factoring quadratics.
• D. (x - 5)(x + 20): Expanding this expression gives us x² + 15x - 100, which is again not equal to x² - 100. This option highlights the importance of accurate factorization and expansion.

The errors in these options stem from either misapplying the difference of squares pattern or attempting to factor the expression using incorrect combinations of numbers. The correct factorization, (x - 10)(x + 10), stands out as the only one that accurately represents the factors of x² - 100.

Conclusion: The Correct Factors and Key Takeaways

In conclusion, the correct factors of the expression x² - 100 are (x - 10) and (x + 10), as represented in option B. This factorization is achieved by recognizing and applying the difference of squares pattern, a powerful tool in algebraic manipulation. Throughout this exploration, we have emphasized the importance of:

• Understanding the basics of factoring: Factoring is a fundamental skill in algebra that allows us to break down expressions into their constituent parts.
• Recognizing patterns: The difference of squares pattern is a valuable shortcut that simplifies factoring in specific cases.
• Applying the difference of squares: The formula a² - b² = (a - b)(a + b) provides a direct route to factoring expressions in this form.
• Verifying the factors: Expanding the factored expression ensures the accuracy of the solution.
• Avoiding common mistakes: Understanding why incorrect options fail helps solidify the correct approach.

By mastering these concepts and techniques, you can confidently tackle a wide range of factoring problems and enhance your overall algebraic proficiency. Remember, practice is key to success in mathematics, so continue to explore and apply these principles to various expressions and equations. The ability to factor expressions like x² - 100 is not just a mathematical skill; it's a gateway to deeper understanding and problem-solving in various fields of science, engineering, and beyond.