Final Answer:
The expression (a+b)(a-b) represents the difference of perfect squares, resulting in the simplified form a² - b².
Step-by-step explanation:
The given expression (a+b)(a-b) is a product of two binomials, and it falls under a special algebraic case known as the difference of perfect squares. The pattern (a+b)(a-b) can be recognized as (a)² - (b)², where 'a' and 'b' are variables or constants. This algebraic identity can be proven by expanding the expression (a+b)(a-b) using the distributive property. Expanding, we get a² - ab + ab - b². The middle terms (-ab and +ab) cancel each other out, leaving us with a² - b².
Understanding the difference of perfect squares is crucial in algebra as it provides a concise and simplified form for certain polynomial expressions. This identity is particularly useful for factoring and simplifying expressions in mathematics. For example, if you encounter the expression x² - 9, you can recognize it as (x+3)(x-3), applying the difference of perfect squares pattern.
In conclusion, recognizing and utilizing the difference of perfect squares pattern can simplify algebraic expressions, making them easier to work with and understand. This mathematical identity has broad applications in algebraic manipulations, equation solving, and polynomial factorization, showcasing its significance in various mathematical contexts.