<div class="ex">    
Soit \( \displaystyle f(x) = x^3-4x+1  \). On vrifie que \( f \)
admet \( 3 \) racines relles\ \ \(  l_1\in \lbrack -2.5,-2\rbrack \;  \)
\(  l_2\in \lbrack 0,0.5\rbrack \;  \) et \(  \;  l_3\in \lbrack 1.5,2\rbrack  \) en posant 
<div class="math">\( g(x) = x-{x^3-4x+1\over 3x^2-4} = {2x^3-1\over 3x^2-4}\)</div> 
Un simple calcul donne les valeurs suivantes:

 <div class="center">
<table border=1 align="center" class="tableau"><tr><td>

\( x_0 \)&nbsp;</td><td>&nbsp;-2&nbsp;</td><td>&nbsp;0&nbsp;</td><td>&nbsp;2</td></tr><tr><td>

\( x_1 \)&nbsp;</td><td>&nbsp;-2.125&nbsp;</td><td>&nbsp;0.25&nbsp;</td><td>&nbsp;1.875</td></tr><tr><td>
 
\( x_2 \)&nbsp;</td><td>&nbsp;-2.114975450&nbsp;</td><td>&nbsp;0.254098301&nbsp;</td><td>&nbsp;1.860978520</td></tr><tr><td>

\( x_3 \)&nbsp;</td><td>&nbsp;-2.114907545&nbsp;</td><td>&nbsp;0.254101688&nbsp;</td><td>&nbsp;1.860805877</td></tr><tr><td>

\( x_4 \)&nbsp;</td><td>&nbsp;-2.114907541&nbsp;</td><td>&nbsp;0.254101688 &nbsp;</td><td>&nbsp;1.860805853</td></tr><tr><td>

\( x_5 \)&nbsp;</td><td>&nbsp;-2.114907541&nbsp;</td><td>&nbsp;0.254101688 &nbsp;</td><td>&nbsp;1.860805853</td></tr><tr><td>

\( x_6 \)&nbsp;</td><td>&nbsp; &nbsp;</td><td>&nbsp; &nbsp;</td><td>&nbsp; </td></tr><tr><td>

\( x_7 \)&nbsp;</td><td>&nbsp; &nbsp;</td><td>&nbsp; &nbsp;</td><td>&nbsp; </td></tr><tr><td>

\( x_8 \)&nbsp;</td><td>&nbsp; &nbsp;</td><td>&nbsp; &nbsp;</td><td>&nbsp; </td></tr></table>
</div> 

\noindent On constate que les valeurs nuriques se stabilisent et on a alors les valeurs approches de \(  l_1, \)\ \( l_2 \) et \( l_3 \)
 environ \( \displaystyle 10^{-9} \) prs.
</div>