(x-2)(x-3)(x-6)(x+1)

4 min read Jun 17, 2024

Exploring the Polynomial (x-2)(x-3)(x-6)(x+1)

This article delves into the fascinating world of the polynomial expression: (x-2)(x-3)(x-6)(x+1). We'll uncover its key properties, explore its graphical representation, and discover its significance in various mathematical contexts.

Expanding the Expression

Before we delve deeper, let's first expand the expression to gain a clearer understanding:

(x-2)(x-3)(x-6)(x+1) = x⁴ - 10x³ + 17x² + 36x - 36

This expanded form reveals that our polynomial is a fourth-degree polynomial.

Finding the Roots

The roots of a polynomial are the values of x that make the expression equal to zero. For our polynomial, we can easily identify the roots by observing the factored form:

x = 2
x = 3
x = 6
x = -1

These are the x-intercepts of the polynomial's graph, representing the points where the graph crosses the x-axis.

Graphing the Polynomial

The graph of (x-2)(x-3)(x-6)(x+1) showcases its behavior and characteristics:

Shape: The graph is a curve that rises to the left and rises to the right. This is because it's a fourth-degree polynomial with a positive leading coefficient.
Intercepts: We know the x-intercepts are at x = 2, 3, 6, and -1. The y-intercept can be found by setting x = 0 and evaluating the expression, resulting in y = -36.
Turning Points: The graph has three turning points, corresponding to the points where the curve changes from increasing to decreasing or vice-versa.

Applications and Significance

This seemingly simple polynomial has applications in various fields:

Modeling Real-World Phenomena: Polynomials are powerful tools for modeling real-world data, such as population growth, financial trends, and physical processes.
Solving Equations: Understanding polynomial expressions is crucial for solving complex equations in various fields like physics and engineering.
Calculus: Derivatives and integrals of polynomials play a significant role in calculus, used for analyzing rates of change and areas under curves.

Conclusion

The polynomial (x-2)(x-3)(x-6)(x+1) serves as a fascinating example of how a seemingly simple expression can reveal rich mathematical properties. By understanding its roots, graph, and applications, we can gain a deeper appreciation for the power and versatility of polynomials in mathematics and beyond.