%2F(x%5E2-1).jpeg "(x^3-1)/(x^2-1)")
Simplifying the Expression (x^3 - 1) / (x^2 - 1)
The expression (x^3 - 1) / (x^2 - 1) can be simplified using factorization and cancellation. Let's break down the steps:
Factorization
1. Difference of Cubes: The numerator (x^3 - 1) is a difference of cubes, which factors into: (a^3 - b^3) = (a - b)(a^2 + ab + b^2)
In this case, a = x and b = 1, so: (x^3 - 1) = (x - 1)(x^2 + x + 1)
2. Difference of Squares: The denominator (x^2 - 1) is a difference of squares, which factors into: (a^2 - b^2) = (a + b)(a - b)
In this case, a = x and b = 1, so: (x^2 - 1) = (x + 1)(x - 1)
Cancellation
Now we have: (x - 1)(x^2 + x + 1) / (x + 1)(x - 1)
Notice that (x - 1) appears in both the numerator and denominator. We can cancel these terms, leaving us with:
(x^2 + x + 1) / (x + 1)
Restrictions
It's important to note that the original expression is undefined when the denominator is zero. This occurs when x = 1 and x = -1. Therefore, the simplified expression (x^2 + x + 1) / (x + 1) is valid for all values of x except for x = -1 and x = 1.
Conclusion
By factoring the numerator and denominator and cancelling common factors, we simplified the expression (x^3 - 1) / (x^2 - 1) to (x^2 + x + 1) / (x + 1), valid for all x except x = -1 and x = 1.