Prove polynomial identities and use them to solve or describe numerical relationships.
Simplify a rational identity while preserving original-domain exclusions
Problem
Simplify \((x^2~-~9)/(x~-~3)\) and state the restriction from the original denominator.
Big Picture
What this problem is really about
Start with the domain, not the cancellation. Find every input that makes the original denominator zero and record those exclusions before factoring. Then factor numerator and denominator, cancel only complete common factors for allowed inputs, and carry the original exclusions into the simplified result, because cancellation changes the formula’s appearance but never restores a missing domain point.
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