76

<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> </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> <bR> </td> <td valign="top" height=432 BGCOLOR="#FFFFFF" TEXT="#080000"> <div> <B>76</B></div> <div> <B>Corolario del Teorema del residuo</B></div> <bR> <div>  <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js"></script>  <ins class="adsbygoogle" style="display:inline-block;width:970px;height:250px" data-ad-client="ca-pub-1578463498418701" data-ad-slot="8553279920"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script>  </div> <bR> <div> <B>P r o c e d i m i e n t o</B></div> <div>      <FONT SIZE="4" COLOR="#006633" FACE="Times New Roman" ><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="Times New Roman" ><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="#006633" FACE="Times New Roman" ><I>Corolario</I></FONT>.</div> <div> <FONT SIZE="4" COLOR="#FF9900" FACE="Times New Roman" ><I><B>Nota2:</B></I></FONT> Si el divisor tiene la forma <I>x - a</I>, entonces para aplicar el <FONT SIZE="4" COLOR="#006633" FACE="Times New Roman" ><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</div> <div>  <I>x - a</I>.  <I> </I></div> <DIV ALIGN="LEFT"></DIV> <div> <B>Enunciados y enlaces a las soluciones de los problemas del Ejercicio 76</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 76:</FONT></div> <bR> <div> <B>H</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >allar, sin efectuar la división, si son exactas o no las divisiones siguientes:</FONT></div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><IMG SRC="78f4c5170.gif" border=0 width="197" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/fs5xUqwPf5U" TARGET="_top" TITLE="https://youtu.be/fs5xUqwPf5U"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >       <IMG SRC="78f5fd170.gif" border=0 width="253" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/fs5xUqwPf5U" TARGET="_top" TITLE="https://youtu.be/fs5xUqwPf5U"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >       <IMG SRC="78f630170.gif" border=0 width="304" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><A HREF="https://youtu.be/fs5xUqwPf5U" TARGET="_top" TITLE="https://youtu.be/fs5xUqwPf5U"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="78f713170.gif" border=0 width="275" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/fs5xUqwPf5U" TARGET="_top" TITLE="https://youtu.be/fs5xUqwPf5U"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="78f808170.gif" border=0 width="264" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/dtcB6KRZ7o0" TARGET="_top" TITLE="https://youtu.be/dtcB6KRZ7o0"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="78f950170.gif" border=0 width="336" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/dtcB6KRZ7o0" TARGET="_top" TITLE="https://youtu.be/dtcB6KRZ7o0"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <B>S</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >in efectuar la división, probar que:</FONT></div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><IMG SRC="78fa48170.gif" border=0 width="328" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="#7" TITLE="76"><U><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="https://youtu.be/0gUyrmVOnvY" TARGET="_top" TITLE="https://youtu.be/0gUyrmVOnvY"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >       <IMG SRC="78fb89170.gif" border=0 width="393" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="#8" TITLE="76"><U><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="https://youtu.be/0gUyrmVOnvY" TARGET="_top" TITLE="https://youtu.be/0gUyrmVOnvY"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="78fc69170.gif" border=0 width="361" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/0gUyrmVOnvY" TARGET="_top" TITLE="https://youtu.be/0gUyrmVOnvY"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="78fddb170.gif" border=0 width="475" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/wUR8IoNZaaA" TARGET="_top" TITLE="https://youtu.be/wUR8IoNZaaA"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <B>S</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >in efectuar la división, hallar si las divisiones siguientes son o no exactas, y determinar el cociente en cada caso y el residuo, si lo hay:</FONT></div> <div> <FONT SIZE="4" COLOR="#000000" FACE="Georgia" ><IMG SRC="78fe24170.gif" border=0 width="292" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="#11" TITLE="76"><U><IMG SRC="598018180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><A HREF="https://youtu.be/-uOshYIrm2E" TARGET="_top" TITLE="https://youtu.be/-uOshYIrm2E"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >       <IMG SRC="78ff07170.gif" border=0 width="263" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="https://youtu.be/-uOshYIrm2E" TARGET="_top" TITLE="https://youtu.be/-uOshYIrm2E"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >       <IMG SRC="7900df170.gif" border=0 width="223" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></FONT><A HREF="https://youtu.be/-uOshYIrm2E" TARGET="_top" TITLE="https://youtu.be/-uOshYIrm2E"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></div> <div> <IMG SRC="790193170.gif" border=0 width="403" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/-k7GIAehsH0" TARGET="_top" TITLE="https://youtu.be/-k7GIAehsH0"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="79028f170.gif" border=0 width="399" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/-k7GIAehsH0" TARGET="_top" TITLE="https://youtu.be/-k7GIAehsH0"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="790373170.gif" border=0 width="371" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/YVdFluixNz4" TARGET="_top" TITLE="https://youtu.be/YVdFluixNz4"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="7904af170.gif" border=0 width="431" height="23" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/YVdFluixNz4" TARGET="_top" TITLE="https://youtu.be/YVdFluixNz4"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <FONT SIZE="4" COLOR="#0099FF" FACE="Georgia" ><B>E</B></FONT>n los ejemplos siguientes, hallar el valor de la constante <I>k</I> (término independiente del polinomio) para que:</div> <div> <IMG SRC="7905481b0.gif" border=0 width="328" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/P7c-gbG9m8E" TARGET="_top" TITLE="https://youtu.be/P7c-gbG9m8E"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="79066c1b0.gif" border=0 width="364" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/P7c-gbG9m8E" TARGET="_top" TITLE="https://youtu.be/P7c-gbG9m8E"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT></div> <div> <IMG SRC="7907591b0.gif" border=0 width="345" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/P7c-gbG9m8E" TARGET="_top" TITLE="https://youtu.be/P7c-gbG9m8E"><U><IMG SRC="598118180.jpg" border=0 width="24" height="24" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></U></A></FONT>       <IMG SRC="7908951b0.gif" border=0 width="405" height="27" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"><FONT SIZE="4" COLOR="#000000" FACE="Bell MT" ><A HREF="https://youtu.be/P7c-gbG9m8E" TARGET="_top" TITLE="https://youtu.be/P7c-gbG9m8E"><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>Lista de reproducción del Ejercicio 76</B></div> <bR> <div> <B> <iframe width="560" height="315" src="https://www.youtube.com/embed/videoseries?list=PLq_SQwoOZHo1HAo4UXhqcmJncVIBowuLc" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>  </B></div> <bR> <div> <B>H</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >allar, sin efectuar la división, si son exactas o no las divisiones siguientes:</FONT></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="5c96d9d70.gif" border=0 width="985" height="215" ALT="MathType 6.0 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="78f300d70.gif" border=0 width="1024" height="215" 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="5bdae9c30.gif" border=0 width="489" height="195" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B>S</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >in efectuar la división, probar que:</FONT></div> <div>  <A NAME="7"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>7</div> <div> <IMG SRC="5c985d170.gif" border=0 width="861" height="279" ALT="MathType 6.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <div>  <A NAME="8"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>8</div> <div> <IMG SRC="5bdb01c30.gif" border=0 width="769" height="195" ALT="MathType 5.0 Equation" ALIGN="BOTTOM" HSPACE="0" VSPACE="0"></div> <bR> <div> <B>S</B><FONT SIZE="4" COLOR="#000000" FACE="Georgia" >in efectuar la división, hallar si las divisiones siguientes son o no exactas, y determinar el cociente en cada caso y el residuo, si lo hay:</FONT></div> <div>  <A NAME="11"><IMG BORDER="0" SRC="1x1.gif" HEIGHT="1" ALIGN="bottom" WIDTH="1" HSPACE="0" VSPACE="0" ></A>11</div> <div> <IMG SRC="5bdcd1330.gif" border=0 width="721" height="307" ALT="MathType 5.0 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" 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