Rewrite rational expressions using inspection, polynomial division, or technology.
Simplify a factored rational expression without losing holes
Problem
Simplify \((x^2~-~1)/(x~-~1)\) and state its original-domain restriction.
Big Picture
What this problem is really about
A canceled factor simplifies values only where the original expression was already defined. Record the denominator zero first, factor the numerator completely, and cancel the common factor on the allowed domain. The remaining formula may look polynomial, but the original excluded input survives as a hole and must accompany the simplified rule.
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