(n%2B3).jpeg "(3n+2)(n+3)")
Exploring the Expression (3n + 2)(n + 3)
The expression (3n + 2)(n + 3) represents the product of two binomials. Let's delve into its properties, ways to expand it, and its potential applications.
Expanding the Expression
We can expand the expression using the FOIL method (First, Outer, Inner, Last):
- First: 3n * n = 3n²
- Outer: 3n * 3 = 9n
- Inner: 2 * n = 2n
- Last: 2 * 3 = 6
Adding these terms together, we get the expanded form:
3n² + 9n + 2n + 6
Combining like terms, we arrive at the simplified form:
3n² + 11n + 6
Analyzing the Expression
- Degree: The highest power of the variable 'n' is 2, making this a quadratic expression.
- Coefficients: The coefficients are 3, 11, and 6.
- Constant Term: The constant term is 6.
Applications
This expression could arise in various contexts, including:
- Algebraic Manipulation: In solving equations or simplifying complex expressions, you might encounter (3n + 2)(n + 3).
- Geometric Applications: The expression might represent the area of a rectangle with sides of length (3n + 2) and (n + 3).
- Modeling: It could be used to represent a relationship between two variables, where the product of two linear functions is needed.
Further Exploration
- Factoring: The expanded form 3n² + 11n + 6 can be factored back into the original binomials (3n + 2)(n + 3). This demonstrates the relationship between multiplication and factorization.
- Graphing: The graph of the expression 3n² + 11n + 6 would be a parabola. Analyzing the graph can reveal information about the function's behavior.
- Solving Equations: Setting the expression equal to zero and solving for 'n' would yield the roots or solutions of the equation.
In conclusion, (3n + 2)(n + 3) is a simple yet versatile expression with applications across various mathematical disciplines. Understanding its expansion, properties, and potential uses provides a solid foundation for further exploration and problem-solving.