%2F(x%5E2-1).jpeg "(x^4-1)/(x^2-1)")
Simplifying the Expression: (x^4 - 1) / (x^2 - 1)
This expression involves polynomial division, and it can be simplified by factoring both the numerator and the denominator.
Factoring the Expression
1. Factoring the Numerator:
- The numerator (x^4 - 1) is a difference of squares. We can factor it as: (x^2 + 1)(x^2 - 1)
2. Factoring the Denominator:
- The denominator (x^2 - 1) is also a difference of squares: (x + 1)(x - 1)
Simplifying the Expression
Now, let's substitute these factored expressions back into the original expression:
((x^2 + 1)(x^2 - 1)) / ((x + 1)(x - 1))
Notice that (x^2 - 1) appears in both the numerator and denominator. We can cancel these terms out:
(x^2 + 1) / (x + 1)
Restrictions
It's important to note that this simplified expression is only valid for values of x where the original denominator (x^2 - 1) is not equal to zero.
This means that x cannot be equal to 1 or -1.
Conclusion
The simplified form of the expression (x^4 - 1) / (x^2 - 1) is (x^2 + 1) / (x + 1), with the restriction that x ≠ 1 or -1.