(x^2+x+1)(x^2-x+1)(x^4-x^2+1)(x^8-x^4+1)

3 min read Jun 17, 2024

Exploring the Pattern: (x^2+x+1)(x^2-x+1)(x^4-x^2+1)(x^8-x^4+1)

This expression might seem intimidating at first glance, but it exhibits a fascinating pattern that allows for a more straightforward solution. Let's break it down step by step.

Recognizing the Pattern

Notice that each factor within the expression follows a similar structure:

x^2 + x + 1
x^2 - x + 1
x^4 - x^2 + 1
x^8 - x^4 + 1

Each subsequent factor is essentially the square of the previous factor with the middle term removed. This pattern is crucial to simplifying the expression.

The Power of Difference of Squares

The key to simplifying this expression lies in the difference of squares factorization:

a² - b² = (a + b)(a - b)

We can use this identity to manipulate our factors:

(x² + x + 1)(x² - x + 1) = (x^2 + 1)² - (x)²

Notice that the right side of the equation is now in the form of the difference of squares. Applying this again, we get:

(x^4 + 2x² + 1) - (x)² = (x^4 + x² + 1)(x^4 - x² + 1)

We can continue this process, applying the difference of squares factorization repeatedly:

(x^4 + x² + 1)(x^4 - x² + 1) = (x^8 + 2x^4 + 1) - (x^4)² = (x^8 + x^4 + 1)(x^8 - x^4 + 1)

Now, we've expanded the expression to include the next two factors.

Simplifying the Expression

By continuing this pattern, we can simplify the entire expression:

(x^2 + x + 1)(x^2 - x + 1)(x^4 - x^2 + 1)(x^8 - x^4 + 1) = (x^16 + x^8 + 1)(x^16 - x^8 + 1) = x^32 + x^16 + 1

Conclusion

The expression, which seemed complex at first, can be simplified through the repeated application of the difference of squares factorization. The pattern present in the factors allows for a more elegant solution, demonstrating the power of recognizing mathematical patterns.