<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>2.1.17 (Remainder theorem)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h1>Remainder Theorem</h1>

<p>When the polynomial \(P(x)\) is divided by \(x-a\), the remainder is \(P(a)\), i.e. in the notation above, \(r=P(a)\).</p>

<p><b>Proof</b></p>
<p>We have established that we can write \(P(x) = (x-a)Q(x) +&nbsp; r\) for some&nbsp;\(Q(x)\) and \(r\).</p>
<p>Using this, 
\[
\begin{eqnarray} 
P(a) &amp;=&amp; (a-a)Q(a)+r 
\\ 
&amp;=&amp; 0\times Q(a)+r &nbsp;
\\ 
&amp;=&amp; r,
\end{eqnarray}
\] 
which is what we wanted to show.</p>
<br/>
<p>For the example above, \(P(x)=x^2+x-10\), we worked out that the remainder when dividing by \(x+4\) is \(2\), and indeed \(P(-4)=16-4-10=2\).</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
