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

2 min read Jun 17, 2024
(x^4+1)(x^2+1)(x+1)(x-1)

Factoring and Expanding the Expression (x⁴ + 1)(x² + 1)(x + 1)(x - 1)

This expression presents a fascinating opportunity to explore the interplay between factorization and expansion in algebra. Let's break down the process step-by-step.

Recognizing Key Patterns

  • Difference of Squares: Notice the (x + 1) and (x - 1) terms. These represent the classic difference of squares pattern: (a + b)(a - b) = a² - b²
  • Sum of Squares: The (x⁴ + 1) and (x² + 1) terms might seem less familiar, but they are both irreducible over the real numbers.

Step-by-Step Solution

  1. Initial Expansion: Begin by expanding the difference of squares: (x + 1)(x - 1) = x² - 1

  2. Substitution: Substitute the result into the original expression: (x⁴ + 1)(x² + 1)(x² - 1)

  3. Further Expansion: Now, we can expand the product of the remaining terms: (x⁴ + 1)(x⁴ - 1)

  4. Final Difference of Squares: We encounter another difference of squares: (x⁸ - 1)

The Final Result

The fully expanded form of the expression is x⁸ - 1.

Key Takeaways

  • Strategic Factoring: Identifying patterns like the difference of squares allows for efficient simplification.
  • Irreducible Factors: Some expressions, like (x⁴ + 1) and (x² + 1), may not be factorable over the real numbers.
  • Expansion and Factorization: These two operations are often used in tandem to manipulate expressions and solve problems in algebra.

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