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      <text>10.1.3.3 Solutions of Polynomial equations</text>
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      <text><![CDATA[<h4 class="HELM_example">Example </h4>
<p>Given that <span class="math inline">\(x=3-2i\)</span> is one root of the cubic equation <span class="math inline">\(x^3-7x^2+19x-13=0\)</span>, find the other two roots.</p>

<h4 class="HELM_solution">Solution</h4>
<p>Since the coefficients of the equation are real and <span class="math inline">\(3-2i\)</span> is a root then its complex conjugate <span class="math inline">\(3+2i\)</span> is also a root which implies that <span class="math inline">\(x-(3-2i)\)</span>    and <span class="math inline">\(x-(3+2i)\)</span> are factors of the given cubic expression. Multiplying together these two factors: <span class="math inline">\[(x-(3-2i))(x-(3+2i))=x^2-x(3-2i)-x(3+2i)+13=x^2-6x+13\]</span> so <span class="math inline">\(x^2-6x+13\)</span>    is a quadratic factor of the cubic equation. <br></p>
<p>The remaining factor <span><strong>must</strong></span> take the form <span class="math inline">\((x+a)\)</span> where <span class="math inline">\(a\)</span> is real, since only one more linear factor of the cubic equation is required, and so we write:
    <span class="math inline">\[ x^3-7x^2+19x-13=(x^2-6x+13)(x+a).\]</span> By inspection (consider for example the constant terms), it is clear that <span class="math inline">\(a=-1\)</span> so that the final factor is <span class="math inline">\((x-1)\)</span>,
    implying that the original cubic equation has a root at <span class="math inline">\(x=1\)</span>.</p>]]></text>
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