(6n%2B1).jpeg "(2n+2)(6n+1)")
Exploring the Expression (2n+2)(6n+1)
The expression (2n+2)(6n+1) represents a product of two binomials, each with a variable term and a constant term. We can analyze this expression in several ways:
1. Expanding the Expression
We can expand the expression using the distributive property (also known as FOIL method):
- First: 2n * 6n = 12n²
- Outer: 2n * 1 = 2n
- Inner: 2 * 6n = 12n
- Last: 2 * 1 = 2
Combining the terms, we get the expanded form: 12n² + 14n + 2
2. Factoring the Expression
The expanded form (12n² + 14n + 2) can be factored. We look for two numbers that multiply to 24 (12 * 2) and add up to 14. These numbers are 12 and 2.
Therefore, we can factor the expression as: (2n + 1)(6n + 2)
It's important to note that this factored form can be further simplified by factoring out a 2 from the second binomial: 2(2n+1)(3n+1)
3. Evaluating the Expression
To evaluate the expression for a given value of 'n', we simply substitute the value and perform the calculations.
For example, if n = 2, then: (2n+2)(6n+1) = (22+2)(62+1) = (4+2)(12+1) = 6 * 13 = 78
4. Applications
The expression (2n+2)(6n+1) can be used in various contexts, including:
- Algebraic manipulation: Understanding its expansion and factorization helps simplify more complex equations and expressions.
- Modeling real-world scenarios: This expression might represent the area of a rectangle with sides (2n+2) and (6n+1), or the total cost of buying 'n' items with a price of (2n+2) and 'n' items with a price of (6n+1).
Overall, understanding the expression (2n+2)(6n+1) provides valuable insight into algebraic manipulation, factorization, and its potential applications in various scenarios.