<p>Find the number of integral solutions for <i>n</i> such that <i>(n + i)</i><sup>4</sup> has zero imaginary part.</p>
Step-by-Step Solution
Key Concept: For a complex number to have zero imaginary part, the coefficient of i must equal zero. Expand and equate the imaginary part to zero.
<p><strong>Step 1:</strong> Expand <i>(n + i)</i><sup>4</sup>.</p><p><i>(n + i)</i><sup>2</sup> = <i>n</i><sup>2</sup> - 1 + 2<i>ni</i></p><p><i>(n + i)</i><sup>4</sup> = (<i>n</i><sup>2</sup> - 1 + 2<i>ni</i>)<sup>2</sup> = (<i>n</i><sup>2</sup> - 1)<sup>2</sup> - 4<i>n</i><sup>2</sup> + 4<i>n</i>(<i>n</i><sup>2</sup> - 1)<i>i</i></p><p><strong>Step 2:</strong> For the imaginary part to be zero:</p><p>4<i>n</i>(<i>n</i><sup>2</sup> - 1) = 0</p><p><strong>Step 3:</strong> Solve for <i>n</i>:</p><p><i>n</i> = 0 or <i>n</i> = ±1</p><p>∴ The number of integral solutions is <strong>3</strong>.</p>
Correct Answer: 3