Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
Find a missing coefficient from remainder information using the Remainder Theorem
Problem
The remainder is \(7\) when \(p(x)~=~x^2~+~kx~+~1\) is divided by \(x~-~2\). Find \(k\).
Big Picture
What this problem is really about
The stated remainder gives an equation for the unknown coefficient through the Remainder Theorem. Since the divisor is x minus 2, we’ll set p of 2 equal to 7, substitute carefully so k x becomes 2 k, and solve the resulting linear equation. Checking the recovered coefficient in p of 2 confirms the required remainder.
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