Rewrite rational expressions using inspection, polynomial division, or technology.
Decide rational-rewrite equivalence using algebra and original domains
Problem
Determine whether \(A~=~x~+~2~+~3/(x~+~1)\) and \(B~=~(x^2~+~3x~+~5)/(x~+~1)\) are equivalent. Justify the result algebraically and compare their domains.
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What this problem is really about
Equivalence of rational expressions requires two matches: the same value formula and the same domain. Recombine a quotient-plus-remainder form over its shared denominator, simplify the numerator, and compare it with the other expression. Then list the original denominator zeros on both sides; matching formulas with different exclusions would still describe different functions.
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