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

3 min read Jun 17, 2024

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

This article delves into the cubic polynomial (x-1)(x+2)(x-3), analyzing its properties, graphing its behavior, and exploring its applications.

Expanding the Polynomial

The first step is to expand the factored form to obtain the standard polynomial form:

(x-1)(x+2)(x-3) = (x² + x - 2)(x-3) = x³ - 2x² - 7x + 6

This cubic polynomial has a leading coefficient of 1 and a constant term of 6.

Finding the Roots

The roots of a polynomial are the values of x that make the polynomial equal to zero. Since the polynomial is already in factored form, finding the roots is straightforward:

x - 1 = 0 => x = 1
x + 2 = 0 => x = -2
x - 3 = 0 => x = 3

Therefore, the roots of the polynomial are 1, -2, and 3.

Graphing the Polynomial

The graph of this cubic polynomial will have the following characteristics:

Shape: A cubic polynomial has a general "S" shape.
Intercepts: It will cross the x-axis at the roots (1, -2, and 3).
End Behavior: As x approaches positive or negative infinity, the polynomial will approach positive or negative infinity depending on the sign of the leading coefficient (which is positive in this case).

Using these characteristics, we can sketch a rough graph of the polynomial.

Applications

Cubic polynomials like this one find applications in various fields, including:

Physics: Modeling motion, projectile trajectories, and oscillations.
Engineering: Designing curves, analyzing circuits, and solving structural problems.
Economics: Analyzing market trends and predicting economic growth.

Summary

The polynomial (x-1)(x+2)(x-3) is a cubic function with roots at 1, -2, and 3. Its graph has an "S" shape and crosses the x-axis at the roots. Cubic polynomials are valuable tools used in various scientific and engineering disciplines.