<p>Let <i>w</i> and <i>w²</i> be non real cube roots of unity. The least possible degree of a polynomial with real coefficients having roots <i>2w, (2 + 3w), (2 + 3w)², (2 - w - w²)</i> is:</p>
Step-by-Step Solution
Key Concept: For a polynomial with real coefficients, complex roots must appear in conjugate pairs. Use the constraint that w + w² = -1 to simplify expressions.
<p><strong>Analysis:</strong> Since <i>w</i> is a non-real cube root of unity, a polynomial with real coefficients must have complex conjugate pairs as roots.</p><p>The roots given are <i>2w</i>, <i>(2 + 3w)</i>, <i>(2 + 3w)²</i>, and <i>(2 - w - w²) = 2 - (-1) = 3</i> (since <i>w + w² = -1</i>).</p><p>The root <i>3</i> is real, while the other roots are complex. Each complex root requires its conjugate to be included in the polynomial for real coefficients. Additionally, since <i>w</i> is a non-real cube root of unity, conjugate roots related to <i>w</i> and <i>w²</i> must both be included.</p><p>This gives us a minimum of <b>4</b> roots, making the least degree <b>4</b>.</p>
Correct Answer: S