Exploring the Polynomial (x+1)(x2)(2x3)
This article explores the polynomial expression (x+1)(x2)(2x3), delving into its properties, factorization, and potential applications.
Understanding the Expression
The expression (x+1)(x2)(2x3) represents a polynomial in its factored form. It's a product of three linear factors:
 (x+1): This factor indicates a root of the polynomial at x = 1.
 (x2): This factor indicates a root of the polynomial at x = 2.
 (2x3): This factor indicates a root of the polynomial at x = 3/2.
Expanding the Expression
To understand the polynomial's behavior more thoroughly, we can expand the expression by multiplying the factors:

Multiply (x+1) and (x2): (x+1)(x2) = x²  x  2

Multiply the result by (2x3): (x²  x  2)(2x3) = 2x³  5x² + x + 6
Therefore, the expanded form of the polynomial is 2x³  5x² + x + 6.
Finding the Roots
We already know the roots from the factored form, but we can also find them by setting the expanded form equal to zero and solving for x:
2x³  5x² + x + 6 = 0
Finding the roots of a cubic equation can be complex. In this case, we can use the factored form to quickly identify the roots:
 x = 1
 x = 2
 x = 3/2
Graphing the Polynomial
The graph of the polynomial will intersect the xaxis at the roots we just calculated. It's a cubic function, so it will have a general Sshape with a maximum and a minimum point.
Applications
Polynomials like this one have a variety of applications, including:
 Modeling realworld phenomena: They can be used to represent relationships between variables in areas like physics, engineering, and economics.
 Solving equations: Finding the roots of a polynomial helps solve equations related to the phenomenon being modeled.
 Data analysis: Polynomials can be used to fit data points and make predictions about future trends.
Summary
The polynomial (x+1)(x2)(2x3), which can be expanded to 2x³  5x² + x + 6, is a cubic function with roots at x = 1, x = 2, and x = 3/2. Understanding its factored form allows us to easily identify its roots and visualize its graph. This polynomial has wide applications in various fields, making it a valuable tool for mathematicians and scientists.