(x%2B1)(x-1).jpeg "(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.