<p>The complex number \(w\) is given by \(1 + 2\mathrm{i}\).</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find \(w^3\) in the form \(x + \mathrm{i}y\), showing your working.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Given that&nbsp;\(w\) is a root of the equation \(az^3 + 5z^2 + 17z + b = 0\), find the values of the real numbers \(a\) and \(b\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Using these values of \(a\) and \(b\), find all the roots of this equation in exact form.<span class="inline-marks">[3]</span></div></li></ol>

The complex number \(w\) is given by \(1 + 2\mathrm{i}\).

Find \(w^3\) in the form \(x + \mathrm{i}y\), showing your working.

Given that \(w\) is a root of the equation \(az^3 + 5z^2 + 17z + b = 0\), find the values of the real numbers \(a\) and \(b\).

Using these values of \(a\) and \(b\), find all the roots of this equation in exact form.

N2013 P1 Q4

N2013 P1 Q4