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    <name>
      <text>3.3.8.1 Solving polynomial equations graphically</text>
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    <questiontext format="html">
      <text><![CDATA[<h3>
    <p>Solving polynomial equations graphically</p>
</h3>
<p>Polynomial equations, particularly of high degree, are difficult to solve unless they take a particularly simple form. A useful guide to the approximate values of the solutions can be obtained by sketching the polynomial, and discovering where the curve
    crosses the \(x\)-axis. The real roots of the polynomial equation \(P(x)=0\) are given by the values of the intercepts of the function \(y=P(x)\) with the \(x\)-axis because on the \(x\)-axis \(y=P(x)\) is zero. Computer software packages and graphics
    calculators exist which can be used for plotting graphs and hence for solving polynomial equations approximately. Suppose the graph of \(y=P(x)\) is plotted and takes a form similar to that shown below.</p>
<figure class="HELM_figure">
  <img src="@@PLUGINFILE@@/block3fig1-1.svg" alt="block3fig1" width="208" height="151" class="img-responsive atto_image_button_text-bottom">
<figcaption> A polynomial function which cuts the \(x\)-axis at points \(x_1\), \(x_2\), and \(x_3\).</figcaption>
</figure>
<p>The graph intersects the \(x\)-axis at \(x=x_1\), \(x=x_2\), and \(x=x_3\) and so the equation \(P(x)=0\) has three roots \(x_1\), \(x_2\), and \(x_3\), because \(P(x_1)=0\), \(P(x_2)=0\), and \(P(x_3)=0\).</p>

<hr>
<h4 class="HELM_example">Example</h4>
<p></p>
<p>Plot a graph of the function \(y=4x^4-15x^2+5x+6\) and hence approximately solve the equation \(4x^4-15x^2+5x+6 = 0\).</p>
<p></p>
<h4 class="HELM_solution">Solution</h4>
<p></p>
<p>The graph has been plotted here with the aid of a computer graph plotting package and is shown in the figure below. By hand, a less accurate result would be produced, of course.</p>
<p></p>
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    <img src="@@PLUGINFILE@@/block3fig2-1.svg" alt="block3fig2" width="224" height="160" class="img-responsive atto_image_button_text-bottom">

    <figcaption>Graph of \(y=4x^4-15x^2+5x+6\).</figcaption>
</figure>
<p>The solutions of the equation are found by looking for where the graph crosses the horizontal axis. Careful examination shows the solutions are at or close to \(x=1\), \(x=1.5\), \(x=-0.5\), \(x=-2\).</p>
<p></p>
<hr>
<p></p>
<p>An important feature of the graph of a polynomial is that it is <b>continuous</b>. There are never any gaps or jumps in the curve. Polynomial curves never turn back on themselves in the horizontal direction, (unlike a circle). By studying the first graph
    in this section you will see that if we choose any two values of \(x\), say \(a\) and \(b\), such that \(y(a)\) and \(y(b)\) have opposite signs, then at least one root lies between \(x=a\) and \(x=b\).<br></p>
<hr>
<p></p>
<p></p>
<p><i>Try for yourself</i>:&nbsp;Consider the polynomial \(P(x)=5x^3-47x^2+84x\). By evaluating \(P(2)\) and \(P(3)\) show that at least one root of \(P(x)=0\) lies between \(x=2\) and \(x=3\).<br></p>
<hr>
<p></p><i>Try for yourself</i>: Without solving the equation or using a graphical calculator, show that \(x^4+4x-1=0\) has a root between \(x=0\) and \(x=1\).
<p></p>
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