Rewrite rational expressions using inspection, polynomial division, or technology.
Verify a rational-expression rewrite using quotient and remainder
Problem
Verify \((x^2~+~3x~+~5)/(x~+~1)~=~x~+~2~+~3/(x~+~1)\) by multiplying back and stating the domain restriction.
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What this problem is really about
Verify a rational rewrite by reversing the division. Identify the divisor, quotient, and remainder, multiply divisor by quotient, and then add the remainder; the result must equal the original numerator exactly. Separately solve the original denominator’s zero condition, because matching algebraic formulas is not enough unless the rewrite also preserves the original domain.
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