(x-1/x)(x+1/x)(x^2+1/x^2)(x^4+1/x^4)

3 min read Jun 17, 2024

Simplifying the Expression (x-1/x)(x+1/x)(x^2+1/x^2)(x^4+1/x^4)

This expression looks complex, but we can simplify it using a clever pattern and algebraic manipulations. Let's break it down step by step:

Recognizing the Pattern

The expression consists of a series of factors, each of the form (x + 1/x) where the power of 'x' increases in each factor. This pattern suggests a strategy using difference of squares factorization.

Applying Difference of Squares

Let's start by simplifying the first two factors:

(x - 1/x)(x + 1/x) = x² - (1/x)² (Using the difference of squares formula: a² - b² = (a+b)(a-b))
x² - (1/x)² = x² - 1/x²

Now, let's consider the next factor:

(x² + 1/x²) = (x² - 1/x²) + (2/x²) (Adding and subtracting 2/x²)

Notice that (x² - 1/x²) is the result of our previous simplification. This suggests we can use this approach to simplify the entire expression.

Repeating the Process

Let's apply the same logic to the remaining factors:

(x⁴ + 1/x⁴) = (x⁴ - 1/x⁴) + (2/x⁴)
(x⁴ - 1/x⁴) = (x² + 1/x²)(x² - 1/x²) (Difference of squares again!)

Putting it All Together

Now we can substitute and simplify the entire expression:

(x - 1/x)(x + 1/x)(x² + 1/x²)(x⁴ + 1/x⁴) = (x² - 1/x²) * [(x² - 1/x²) + (2/x²)] * [(x² + 1/x²)(x² - 1/x²) + (2/x⁴)] = (x² - 1/x²)² * [(x² - 1/x²) + (2/x²)] + (x² - 1/x²)² * (2/x⁴) = (x² - 1/x²)² * [x² - 1/x² + 2/x² + 2/x⁴] = (x² - 1/x²)² * [x² + 1/x⁴]

Final Simplification

The final expression can be further simplified if desired, but it's already significantly less complex than the original. We have expressed the initial complicated product in terms of squares and simple additions. This demonstrates the power of algebraic techniques to simplify complex expressions.