<p>The complex number \(\omega\), where&nbsp;\(\omega \neq 1\), is a root of the equation \(z^3 - 1 = 0\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(1 + \omega + \omega^2 = 0\).</div></li><li><div class="list-item-paragraph">Hence show that&nbsp;\(1 + \omega^k + \omega^{2k}\), where&nbsp;\(k\) is a positive integer, is not zero when&nbsp;\(k\) is a multiple of \(3\), but zero when&nbsp;\(k\) is not a multiple of \(3\).</div></li></ol>

The complex number \(\omega\), where \(\omega \neq 1\), is a root of the equation \(z^3 - 1 = 0\).

Hence show that \(1 + \omega^k + \omega^{2k}\), where \(k\) is a positive integer, is not zero when \(k\) is a multiple of \(3\), but zero when \(k\) is not a multiple of \(3\).

Special Results involving Exponential Forms [Out of Syllabus]

Complex Numbers

International Baccalaureate Higher Level Mathematics

Geometric sum of roots of unity

Geometric sum of roots of unity