(x+1)(x-2)(2x-3)

4 min read Jun 16, 2024

Exploring the Polynomial (x+1)(x-2)(2x-3)

This article explores the polynomial expression (x+1)(x-2)(2x-3), delving into its properties, factorization, and potential applications.

Understanding the Expression

The expression (x+1)(x-2)(2x-3) 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.
(x-2): This factor indicates a root of the polynomial at x = 2.
(2x-3): 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 (x-2): (x+1)(x-2) = x² - x - 2
Multiply the result by (2x-3): (x² - x - 2)(2x-3) = 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 x-axis at the roots we just calculated. It's a cubic function, so it will have a general S-shape with a maximum and a minimum point.

Applications

Polynomials like this one have a variety of applications, including:

Modeling real-world 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)(x-2)(2x-3), 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.