Rewrite rational expressions using inspection, polynomial division, or technology.
Find a missing coefficient using the remainder when dividing by a linear binomial
Problem
Find \(k\) so that \(x^2~+~kx~+~5\) has remainder \(2\) when divided by \(x~+~1\).
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What this problem is really about
The Remainder Theorem turns a division condition into one evaluation equation. Rewrite the divisor as x minus a to identify the correct signed input, set the polynomial’s value at that input equal to the stated remainder, and solve for the unknown coefficient. Substitute the result back into the same evaluation to check both the sign and the remainder.
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