<p>Let <em>P(n)</em> = x<sup>n</sup> − 1 is divisible by x − k. The least integral value of k is:</p>
Step-by-Step Solution
Key Concept: Use the Factor Theorem: (x - k) divides (x^n - 1) if and only if k^n = 1. For integral k, this means k ∈ {-1, 1}, and we need the least value.
<p><strong>Step 1:</strong> By the Factor Theorem, (x - k) divides P(x) = x^n - 1 if and only if P(k) = 0.</p><p><strong>Step 2:</strong> This means k^n - 1 = 0, so k^n = 1.</p><p><strong>Step 3:</strong> For integral values of k, we need k^n = 1. The only integral solutions are k = 1 and k = -1 (since (-1)^n = 1 when n is even).</p><p><strong>Step 4:</strong> The least integral value between {-1, 1} is k = -1.</p><p>∴ Answer: B (k = -1)</p>
Correct Answer: B