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    <name>
      <text>2.7.1.1</text>
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      <text><![CDATA[<h4>Polynomial functions</h4>

<p>The class of the <b>polynomial functions</b> plays a very important role. Polynomial functions are made up of multiples of non-negative whole number powers of a variable, such as \(3x^2\), \(7x^3\) and so on. You are already familiar with many such functions. Other examples include:</p>
<ul><li>\(P_0(t)=6\),<br></li>
<li>\(P_1(t) = 3t+ 9\),<br></li>
<li>\(P_2(x) = 3x^2−x+ 2\),<br></li>
<li>\(P_4(z) = 7z^4+z^2−1\),<br></li>
</ul>
<p>where \(t\), \(x\) and \(z\) are independent variables.</p><br>
<p> If the powers of the independent variable are fractional or negative, then the function does not represent a polynomial; i.e. \(f(x)=x^{-1}\) and \(g(x)=x^{3/2}\) are not polynomials. The function \(P_0(t)=6\) is a polynomial instead, in fact we can regard it as \(P_0(t)=6 t^0\).</p>

<p>By convention, a polynomial is written with the powers either increasing or decreasing. For example the polynomial</p>
<div style="margin-left: 30px;" class="editor-indent"><p>\(3x+ 9x^2−x^3+ 2\)</p></div>
<p>would be written as</p>
<div style="margin-left: 30px;" class="editor-indent"><p>\(−x^3+ 9x^2+ 3x+ 2\)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; or&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; \(2 + 3x+ 9x^2−x^3\).</p></div>
<p>In general, we have the following definition: </p>

<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>A <b>polynomial expression</b> has the form:</p>
<p>\[a_n x^n+a_{n−1} x^{n−1}+a_{n−2} x^{n−2}+...+a_2 x^2+a_1 x+a_0\]</p>
<p>where \(n\) is a non-negative integer, a_n, a_{n−1}, ..., a_1, a_0 are constants and \(x\) is a variable.</p>
<p>A <b>polynomial function</b> \(P(x)\) has the form</p>
<p>\[P_n(x)=a_n x^n+a_{n−1} x^{n−1}+a_{n−2} x^{n−2}+...+a_2 x^2+a_1 x+a_0.\]</p>
</div>
<p>The <b>degree</b> of a polynomial or polynomial function is the value of the highest power. Referring to the examples listed above, the polynomial \(P_2\) has degree \(2\), because the term with the highest power is \(3x^2\), \(P_4\) has degree \(4\), \(P_1\) has degree \(1\) and \(P_0\) has degree \(0\). Polynomials with low degrees have special names given in the following table. We assume that \(a \neq 0\) in each case.</p><p><br></p>
<center>
<table frame="void" rules="all" border="1">
<thead>
<tr class="header">
<th style="text-align: center;"></th>
<th style="text-align: center;">degree<br></th>
<th style="text-align: center;">name<br></th>
</tr></thead>

<tbody>
<tr class="odd">
<td style="text-align: center;"><span class="math inline">\(a\)</span></td>
<td style="text-align: center;">0<br></td>
<td style="text-align: center;">constant<br></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><span class="math inline">\(ax+b\)</span></td>
<td style="text-align: center;">1<br></td>
<td style="text-align: center;">linear<br></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><span class="math inline">\(ax^2+bx+c\)</span></td>
<td style="text-align: center;">2<br></td>
<td style="text-align: center;">quadratic<br></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><span class="math inline">\(ax^3+bx^2+cx+d\)</span></td>
<td style="text-align: center;">3<br></td>
<td style="text-align: center;">cubic<br></td>
</tr>
<tr class="odd">
<td style="text-align: center;"><span class="math inline">\(ax^4+bx^3+cx^2+dx+e\)</span></td>
<td style="text-align: center;">4<br></td>
<td style="text-align: center;">quartic<br></td>
</tr>
</tbody>
</table>
</center>]]></text>
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