(n%2B1).jpeg "(3n-4)(n+1)")
Exploring the Expression (3n - 4)(n + 1)
The expression (3n - 4)(n + 1) represents a product of two binomials. It can be simplified and analyzed in various ways, revealing its properties and applications.
Simplifying the Expression
To simplify the expression, we can use the distributive property (also known as FOIL):
(First) (Outer) (Inner) (Last)
- First: 3n * n = 3n²
- Outer: 3n * 1 = 3n
- Inner: -4 * n = -4n
- Last: -4 * 1 = -4
Combining the terms, we get:
(3n - 4)(n + 1) = 3n² + 3n - 4n - 4
Simplifying further:
(3n - 4)(n + 1) = 3n² - n - 4
Analyzing the Expression
The simplified expression, 3n² - n - 4, represents a quadratic polynomial. This means it has a degree of 2, indicating a parabola when graphed.
Properties of the Expression:
- Leading coefficient: The coefficient of the n² term is 3, indicating that the parabola opens upwards.
- Constant term: The constant term is -4, representing the y-intercept of the parabola.
- Roots: The roots of the polynomial are the values of n that make the expression equal to zero. To find the roots, we can use the quadratic formula or factorization.
Applications of the Expression
The expression (3n - 4)(n + 1) can be used in various contexts, such as:
- Modeling: It can represent a quadratic model for a real-world scenario. For example, it could model the trajectory of a projectile or the growth of a population.
- Equation solving: The expression can be part of an equation to be solved for n.
- Polynomial operations: The expression can be used in operations such as addition, subtraction, multiplication, and division of polynomials.
Conclusion
The expression (3n - 4)(n + 1) is a simple yet versatile mathematical construct. By simplifying and analyzing it, we can gain valuable insights into its properties and potential applications. It serves as a building block for understanding more complex mathematical concepts.