Rewrite rational expressions using inspection, polynomial division, or technology.
Identify the remainder from a rational expression rewrite
Problem
Identify the remainder in \((x^2~+~3x~+~5)/(x~+~1)~=~x~+~2~+~3/(x~+~1)\).
Big Picture
What this problem is really about
Read quotient-plus-remainder form by matching roles, not by comparing sizes. The polynomial part is the quotient, the denominator of the fractional correction is the original divisor, and that fraction’s numerator is the remainder. Reconstructing divisor times quotient plus remainder provides a quick check that the pieces reproduce the original numerator.
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