%2F(x%5E2-1)-simplify.jpeg "(x^3-1)/(x^2-1) Simplify")
Simplifying the Expression (x³-1)/(x²-1)
This expression can be simplified by factoring both the numerator and denominator.
Factoring the Numerator:
The numerator, x³ - 1, is a difference of cubes. It can be factored as follows:
- **a³ - b³ = (a - b)(a² + ab + b²) **
In this case, a = x and b = 1. Applying the formula, we get:
- x³ - 1 = (x - 1)(x² + x + 1)
Factoring the Denominator:
The denominator, x² - 1, is a difference of squares. It can be factored as follows:
- a² - b² = (a + b)(a - b)
In this case, a = x and b = 1. Applying the formula, we get:
- x² - 1 = (x + 1)(x - 1)
Simplifying the Expression:
Now we can rewrite the original expression with the factored numerator and denominator:
- (x³ - 1) / (x² - 1) = (x - 1)(x² + x + 1) / (x + 1)(x - 1)
Notice that we have a common factor of (x - 1) in both the numerator and denominator. We can cancel out this common factor:
- (x - 1)(x² + x + 1) / (x + 1)(x - 1) = (x² + x + 1) / (x + 1)
Therefore, the simplified expression is (x² + x + 1) / (x + 1).
Important Note: This simplified expression is valid for all values of x except x = 1 and x = -1. This is because these values would make the original denominator equal to zero, resulting in an undefined expression.