Factorizing Polynomials: Solving x^3 - 18x^2 108x - 216

Polynomial factorization is a fundamental concept in algebra that involves expressing a given polynomial as a product of simpler polynomials. This process is crucial in simplifying complex expressions, solving equations, and understanding the roots of the polynomial. In this article, we will focus on the factorization of the polynomial x^3 - 18x^2 108x - 216.

Factorization Process

Let's start by examining the given polynomial:

Direct Factorization Technique

To factorize x^3 - 18x^2 108x - 216, we can use several methods. One common approach is to identify a pattern or use known algebraic identities such as the difference of cubes. Here, we can recognize that the expression resembles the difference of cubes formula:

Step 1: Break Down the Polynomial

First, let's rewrite the polynomial in a form that reveals the difference of cubes:

x^3 - 216 - 18x^2 108x

Recognize that 216 can be expressed as 6^3, and the remaining terms can be grouped to form x(x^2 - 36 - 18x):

x(x^2 - 12x - 36) - 18x^2 108x

x(x - 6)(x - 6) - 18x(x - 6)

Notice that x - 6 is a common factor:

(x - 6)(x^2 - 12x - 36)

(x - 6)^3

Alternative Method Using Algebraic Identities

Let's explore an alternative method that utilizes algebraic identities:

Step 1: Identifying Common Terms Using the Sum and Difference of Cubes

Starting from the original polynomial:

x^3 - 18x^2 108x - 216

Recognize that 216 6^3 and apply the formula for the difference of cubes:

(x^3 - 6^3) - 18x(x - 6)

Using the identity a^3 - b^3 (a - b)(a^2 ab b^2), we get:

(x - 6)(x^2 6x 36) - 18x(x - 6)

Notice that x - 6 is a common factor:

(x - 6)[(x^2 6x 36) - 18x]

Further simplify the expression inside the brackets:

(x - 6)[x^2 - 12x 36]

This can be further simplified to:

(x - 6)(x - 6)^2 (x - 6)^3

Conclusion

In conclusion, the polynomial x^3 - 18x^2 108x - 216 can be factorized into (x - 6)^3. This process showcases the importance of recognizing patterns and using algebraic identities to simplify expressions. Understanding these techniques is crucial for solving polynomial equations and analyzing polynomial functions.

For further reading and practice, consider exploring more complex polynomials and experimenting with different factorization methods. This will enhance your algebraic skills and provide a deeper understanding of polynomial functions.