<p>The complex number \(w\) is given by \(1 + 2\mathrm{i}\).</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><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 data-list-item-id="item-ii" data-list-item-path="item-ii"><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 data-list-item-id="item-iii" data-list-item-path="item-iii"><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.

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Solution 1

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