Question:

<div class="solution"><p><strong>Key Idea:</strong> Use the factor theorem: \((x-2)\) is a factor iff \(P(2)=0\).</p><p><strong>Step 1:</strong> <span class="math-block">\[P(2) = 8k + 8k^2 + k^3 = 0\]</p><p><strong>Step 2:</strong> Factor out \(k\): <span class="math-block">\[k(k^2 + 8k + 8) = 0\]Real roots: \(k=0,\ k=-4\pm 2\sqrt{2}\)</p><p><strong>Step 3:</strong> Sum \(= 0 + (-4+2\sqrt{2}) + (-4-2\sqrt{2}) = -8\)</p><p><strong>Answer: \(-8\)</strong></p><div class="trap-box"><strong>Trap:</strong> Do not treat the condition as \(P(x)=0\) for all \(x\). Factor theorem requires only the single substitution \(x=2\).<div class="key-concept"><strong>Key Concept:</strong> Factor theorem + Vieta's formulas for sum of roots