Apply the Remainder Theorem to connect p(a), remainders, and factors x-a.
Find a missing coefficient that makes x-a a factor
Problem
Find \(k\) so that \(x~-~2\) is a factor of \(x^3~+~kx~-~6\).
Big Picture
What this problem is really about
The unknown coefficient is constrained by the requirement that x minus 2 divide the polynomial evenly. We’ll translate that factor statement into p of 2 equals zero, substitute 2 everywhere x appears—including the k x term—and solve the resulting linear equation for k. Substitution back into p of 2 will verify the factor condition.
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