In the realm of mathematics, algebraic expressions often appear complex and daunting. However, with a systematic approach and a solid understanding of fundamental principles, these expressions can be simplified to reveal their underlying structure. This comprehensive guide will delve into the simplification of the expression (a-b)(a+2)-(a+b)(a-2), providing a step-by-step walkthrough that demystifies the process and empowers you to tackle similar algebraic challenges. We will break down each stage, ensuring clarity and comprehension, making this seemingly intricate problem accessible to all.
Expanding the Products: Laying the Foundation for Simplification
The initial step in simplifying this expression involves expanding the products. This means multiplying out the terms within the parentheses using the distributive property, a cornerstone of algebraic manipulation. The distributive property states that a(b+c) = ab + ac, and we will apply this principle diligently to both products in our expression. By expanding the products, we transform the expression from a compact form to a more detailed representation, which allows us to identify and combine like terms.
Let's begin by expanding the first product, (a-b)(a+2). Applying the distributive property, we multiply each term in the first parentheses by each term in the second parentheses. This yields:
(a-b)(a+2) = a(a+2) - b(a+2)
Now, we apply the distributive property again to each term:
a(a+2) = aa + a2 = a² + 2a
-b(a+2) = -ba - b2 = -ab - 2b
Combining these results, we get:
(a-b)(a+2) = a² + 2a - ab - 2b
Next, we expand the second product, (a+b)(a-2), using the same approach:
(a+b)(a-2) = a(a-2) + b(a-2)
Applying the distributive property:
a(a-2) = aa - a2 = a² - 2a
b(a-2) = ba - b2 = ab - 2b
Combining these results, we get:
(a+b)(a-2) = a² - 2a + ab - 2b
Now that we have expanded both products, our expression becomes:
(a² + 2a - ab - 2b) - (a² - 2a + ab - 2b)
This expanded form sets the stage for the next crucial step: combining like terms.
Combining Like Terms: The Art of Algebraic Harmony
With the products expanded, the next step is to combine like terms. Like terms are terms that have the same variable(s) raised to the same power. Identifying and combining like terms is a fundamental technique in algebraic simplification. It allows us to consolidate the expression, reducing its complexity and revealing its essential structure. This process involves careful observation and meticulous application of arithmetic operations.
To begin, let's rewrite our expression, distributing the negative sign in front of the second set of parentheses:
a² + 2a - ab - 2b - a² + 2a - ab + 2b
Now, we can group the like terms together. Let's start with the a² terms:
a² - a²
These terms cancel each other out, resulting in 0. Next, we consider the a terms:
2a + 2a
Combining these terms, we get 4a. Now, let's look at the ab terms:
-ab - ab
Combining these terms, we get -2ab. Finally, we consider the b terms:
-2b + 2b
These terms also cancel each other out, resulting in 0. After combining all the like terms, our expression simplifies to:
4a - 2ab
This simplified expression is significantly less complex than the original, making it easier to work with and interpret. However, we can take this simplification one step further by factoring out the common factor.
Factoring Out the Common Factor: Unveiling the Core Structure
The final step in simplifying our expression is to factor out the common factor. Factoring is the reverse process of expanding, and it involves identifying a factor that is common to all terms in the expression. Factoring can further simplify an expression, revealing its underlying structure and making it easier to analyze and manipulate. In our case, both terms in the simplified expression 4a - 2ab share a common factor.
Observing the terms, we can see that both 4a and -2ab are divisible by 2a. Therefore, 2a is the common factor that we can factor out. To factor out 2a, we divide each term in the expression by 2a and write the expression as a product of 2a and the resulting quotient.
Dividing 4a by 2a, we get:
4a / 2a = 2
Dividing -2ab by 2a, we get:
-2ab / 2a = -b
Therefore, we can rewrite the expression as:
4a - 2ab = 2a(2 - b)
This is the fully simplified form of the expression. By factoring out the common factor, we have revealed the core structure of the expression, making it even more concise and manageable.
Conclusion: Mastering Algebraic Simplification
In this comprehensive guide, we have successfully simplified the expression (a-b)(a+2)-(a+b)(a-2) through a series of systematic steps. We began by expanding the products using the distributive property, transforming the expression into a more detailed form. Next, we combined like terms, consolidating the expression and reducing its complexity. Finally, we factored out the common factor, revealing the core structure of the expression and achieving the most simplified form: 2a(2 - b). This step-by-step approach demonstrates the power of fundamental algebraic principles in tackling complex expressions. By mastering these techniques, you can confidently navigate the world of algebraic simplification and unlock the hidden beauty and elegance of mathematical expressions. Remember, practice is key to success in mathematics. The more you practice simplifying expressions, the more comfortable and confident you will become in your abilities. So, embrace the challenge, and continue to explore the fascinating world of algebra!
