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

4 min read Jun 17, 2024

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

This article delves into the polynomial expression (x-1)(x+1)(x-2), exploring its properties, factorization, and how to find its roots.

Understanding the Expression

The expression (x-1)(x+1)(x-2) represents a product of three linear factors. Each factor corresponds to a specific root of the polynomial:

(x-1): This factor indicates that the polynomial has a root at x=1.
(x+1): This factor indicates that the polynomial has a root at x=-1.
(x-2): This factor indicates that the polynomial has a root at x=2.

Expanding the Expression

To understand the polynomial's behavior and graph, we can expand the expression:

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

This expanded form gives us a cubic polynomial with a leading coefficient of 1.

Finding the Roots

As we mentioned earlier, the roots of the polynomial are the values of x that make the expression equal to zero. We already know these roots from the factored form:

x = 1
x = -1
x = 2

These roots can also be found by setting the expanded form of the polynomial equal to zero and solving the cubic equation.

Graphing the Polynomial

The graph of the polynomial (x-1)(x+1)(x-2) is a cubic curve that intersects the x-axis at the points (1,0), (-1,0), and (2,0). The curve will have a general shape that rises from negative infinity, passes through the root at x=-1, then changes direction and passes through the root at x=1. Finally, it changes direction again and passes through the root at x=2, continuing to rise towards positive infinity.

Applications

This polynomial and its variations can be used in various applications, including:

Modeling real-world phenomena: The polynomial can represent the behavior of systems with three distinct states or stages.
Solving equations: The roots of the polynomial are solutions to the equation (x-1)(x+1)(x-2) = 0.
Calculus: Understanding the polynomial's behavior is important in calculus, especially when analyzing its derivative and integral.

Conclusion

The polynomial (x-1)(x+1)(x-2) is a simple yet informative expression. By understanding its factorization, roots, and graph, we gain insights into its behavior and its potential applications in various fields.