Rewrite rational expressions using inspection, polynomial division, or technology.
Translate polynomial-division technology output into exact algebraic forms
Problem
Use the reported division of \(x^3~+~x^2~+~x~+~1\) by \(x~-~1\) to write the reconstruction identity and rational rewrite.
Big Picture
What this problem is really about
Translate division output in a fixed order. Place the reported quotient coefficients on descending powers, keep the reported remainder separate, and build the reconstruction identity dividend equals divisor times quotient plus remainder. After verifying that identity, divide through by the divisor to obtain the rational rewrite and retain every zero of the original divisor.
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