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

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

Factoring and Expanding (x^2 + 1)(x + 1)(x - 1)

This expression involves a combination of factoring and expanding to simplify it. Let's break it down step by step:

1. Recognizing the Difference of Squares

Notice that the expression (x + 1)(x - 1) is in the form of a difference of squares. This pattern is easily recognized:

  • (a + b)(a - b) = a² - b²

Applying this to our expression:

  • (x + 1)(x - 1) = x² - 1² = x² - 1

2. Simplifying the Expression

Now, our expression becomes:

  • (x² + 1)(x² - 1)

This is again in the form of a difference of squares:

  • (a + b)(a - b) = a² - b²

Where:

  • a = x²
  • b = 1

Applying this:

  • (x² + 1)(x² - 1) = (x²)² - 1² = x⁴ - 1

Final Result

Therefore, the simplified form of the expression (x² + 1)(x + 1)(x - 1) is x⁴ - 1.

Key Points:

  • The difference of squares pattern is a useful tool for simplifying algebraic expressions.
  • Recognizing these patterns can save time and effort in factorization and expansion.
  • Be aware of nested patterns within expressions to apply the appropriate simplification techniques.