77

<body topmargin=0 leftmargin=0 marginheight=0 marginwidth=0> <table cellpadding=0 cellspacing=0 border=0><tr> <td valign="top" colspan=3 height=24 BGCOLOR="#003333" TEXT="#F7F7FF"> <div> <FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>M</B></FONT><FONT SIZE="4" COLOR="#FFFFFF" FACE="Georgia" ><B>atemáticas de la U</B></FONT><B> (MatU)  | </B><A HREF="http://www.jcbmat.com/" TARGET="_top" TITLE="http://www.jcbmat.com/"><U><B>Home</B></U></A><B> | </B><A HREF="index.htm" TARGET="_top" TITLE="Álgebra de Baldor"><U><B>Algebra de Baldor</B></U></A><B> |  </B><A HREF="id1076.htm" TARGET="_top" TITLE="Índice temático"><U><B>Índice temático</B></U></A><B>  |  </B><A HREF="id1073.htm" TARGET="_top" TITLE="Lista"><U><B>Lista de Ejercicios</B></U></A><B> |</B></div> <bR> <div> </div> </td> </tr><tr> <td valign="top" rowspan="2" width=165 BGCOLOR="#FFFFFF" TEXT="#080000"> <bR> <div> <B> <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:160px;height:600px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="3073775660"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </B></div> </td> <td valign="top" height=432 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>77</B></div> <div> <B>Teorema del residuo</B></div> <div> <B><IMG SRC="5bdda7170.gif" border=0 width="423" height="23" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></B></div> <bR> <div>  <script async src="//pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:728px;height:90px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="5388515922"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <B>A p u n t e s </B></div> <div> <FONT SIZE="4" COLOR="#339933" FACE="Bell MT" ><B>Teorema del Residuo:</B></FONT><B> </B>"El residuo de dividir un polinomio entero y racional en <I>x</I> por un binomio de la forma <I>bx - a</I> se obtiene sustituyendo<B>,</B> en el polinomio dado<B>,</B> la <I>x</I> por <I>a/b</I>".</div> <div> <FONT SIZE="4" COLOR="#00CC00" FACE="Bell MT" ><B>Corolario del Teorema del Residuo:</B></FONT>  Un polinomio entero en <I>x</I>, <I>P(x)</I>, que se anula para  <I>x = a/b</I>, o sea que al sustituir la <I>x</I> por <I>a/b</I> en el polinomio el resultado es cero, esto es <I>P(a/b) </I>= 0, es divisible por <I>bx - a</I>.</div> <div> <FONT SIZE="4" COLOR="#FF9900" FACE="Bell MT" ><I><B>Nota1:</B></I></FONT> Se dice que una cantidad es divisible por otra cantidad si al dividir a la primera por la segunda el residuo es cero. <I>El teorema del residuo</I> establece que para hallar el resto de la división de un polinomio entero en <I>x</I> por un binomio de la forma <I>bx - a</I>, sin efectuar la división, basta con sustituir la <I>x</I> por <I>a/b</I>. Conjugando los dos conceptos anteriores se deduce la veracidad del <FONT SIZE="4" COLOR="#00CC00" FACE="Bell MT" ><I>Corolario</I></FONT>.</div> <div> <FONT SIZE="4" COLOR="#FF9900" FACE="Bell MT" ><I><B>Nota2:</B></I></FONT> Si el divisor tiene la forma <I>x - a</I>, entonces para aplicar el <FONT SIZE="4" COLOR="#00CC00" FACE="Bell MT" ><I>Corolario</I></FONT> se halla <I>P</I>(<I>a</I>) y, si <I>P</I>(<I>a</I>) = 0, se concluye que <I>P</I>(<I>x</I>) es divisible por <I>x - a</I>. </div> <div> <IMG SRC="79099d6f0.gif" border=0 width="413" height="111" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <B>Enunciados y enlaces a las soluciones de los problemas del Ejercicio 77</B></div> <bR> <div> <B>S</B><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" >olución en imagen <IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> y o video <IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> de los doce problemas del Ejercicio 77:</FONT></div> <bR> <div> <B>D</B><FONT SIZE="4" COLOR="#000000" FACE="Times New Roman" >iga, por simple inspección, si son exactas las divisiones siguientes y en caso negativo, diga cuál es el residuo:</FONT></div> <div> <IMG SRC="790a4d300.gif" border=0 width="77" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#1" TITLE="77"><U><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="790b5b300.gif" border=0 width="91" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"> <FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#2" TITLE="77"><U><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="790c4f300.gif" border=0 width="79" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#3" TITLE="77"><U><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="790d54300.gif" border=0 width="84" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="#4" TITLE="77"><U><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="790e5b300.gif" border=0 width="91" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/JTz1CK47Pmg" TARGET="_top" TITLE="https://youtu.be/JTz1CK47Pmg"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="790f4f300.gif" border=0 width="79" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/JTz1CK47Pmg" TARGET="_top" TITLE="https://youtu.be/JTz1CK47Pmg"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="791050300.gif" border=0 width="80" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/JTz1CK47Pmg" TARGET="_top" TITLE="https://youtu.be/JTz1CK47Pmg"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="79115b300.gif" border=0 width="91" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/JTz1CK47Pmg" TARGET="_top" TITLE="https://youtu.be/JTz1CK47Pmg"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="79125c300.gif" border=0 width="92" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/oKymXWgHBkk" TARGET="_top" TITLE="https://youtu.be/oKymXWgHBkk"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>        <IMG SRC="79166f300.gif" border=0 width="111" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/oKymXWgHBkk" TARGET="_top" TITLE="https://youtu.be/oKymXWgHBkk"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="791789300.gif" border=0 width="137" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/oKymXWgHBkk" TARGET="_top" TITLE="https://youtu.be/oKymXWgHBkk"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>        <IMG SRC="791875300.gif" border=0 width="117" height="48" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/oKymXWgHBkk" TARGET="_top" TITLE="https://youtu.be/oKymXWgHBkk"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <B>Soluciones de los problemas del Ejercicio 77</B></div> <bR> <div> <B> <iframe width="560" height="315" src="https://www.youtube.com/embed/videoseries?list=PLq_SQwoOZHo0S0vLV9lXQkqURh-gNqSim" frameborder="0" allowfullscreen></iframe>  </B></div> <div>  <A NAME="1"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>1</div> <div> <IMG SRC="5bdee1f00.gif" border=0 width="481" height="240" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="2"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>2</div> <div> <IMG SRC="5bf0e1f00.gif" border=0 width="481" height="240" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="3"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>3</div> <div> <IMG SRC="5bf1bfbf0.gif" border=0 width="447" height="191" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="4"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>4</div> <div> <IMG SRC="5bf2e1f00.gif" border=0 width="481" height="240" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <DIV ALIGN="LEFT"></DIV> <div> <A HREF="http://twitter.com/Calculo21" TARGET="_top" TITLE="http://twitter.com/Calculo21"><U><IMG SRC="2fc828280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="http://www.facebook.com/juan.beltran.58511" TARGET="_top" TITLE="http://www.facebook.com/juan.beltran.585"><U><IMG SRC="76fa28280.gif" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="https://www.youtube.com/user/MatematicaAplicadas" TARGET="_top" TITLE="https://www.youtube.com/user/MatematicaA"><U><IMG SRC="2fca28280.jpg" border=0 width="40" height="40" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <bR> <div> Por: <FONT SIZE="4" COLOR="#FF9900" FACE="Times New Roman" >Juan Carlos </FONT><FONT SIZE="4" COLOR="#0000FF" FACE="Times New Roman" >Beltrán</FONT> <FONT SIZE="4" COLOR="#FF0000" FACE="Times New Roman" >Beltrán </FONT></div> <bR> <div>  <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" 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