(2a-5).jpeg "(2a+5)(2a-5)")
Understanding (2a + 5)(2a - 5)
This expression represents the product of two binomials: (2a + 5) and (2a - 5). To understand this product, let's delve into the concepts involved:
Binomials
Binomials are algebraic expressions consisting of two terms. In our case, both (2a + 5) and (2a - 5) are binomials.
Expanding the Product
To expand the product (2a + 5)(2a - 5), we can apply the distributive property, also known as FOIL (First, Outer, Inner, Last):
- First: Multiply the first terms of each binomial: (2a * 2a) = 4a²
- Outer: Multiply the outer terms: (2a * -5) = -10a
- Inner: Multiply the inner terms: (5 * 2a) = 10a
- Last: Multiply the last terms: (5 * -5) = -25
Combining the results, we get: 4a² - 10a + 10a - 25
Simplifying the Expression
Notice that the terms -10a and +10a cancel each other out. Therefore, the simplified expression is: 4a² - 25
Recognizing the Pattern
The expression (2a + 5)(2a - 5) is a special case called the difference of squares. This pattern is frequently encountered in algebra and can be generalized as:
(a + b)(a - b) = a² - b²
In our example, a = 2a and b = 5.
Importance of the Difference of Squares
The difference of squares pattern is important because it provides a shortcut for factoring certain expressions. By recognizing this pattern, you can quickly factor expressions like 4a² - 25 into (2a + 5)(2a - 5).
Conclusion
Understanding the product (2a + 5)(2a - 5) involves recognizing binomials, applying the distributive property, and simplifying the resulting expression. By mastering this concept, you'll gain valuable insights into algebraic manipulations and the importance of pattern recognition.