document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=312825>Question 312825</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve x^3 - x^2 - 7x +15 =0 over the set of complex numbers </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #223674 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=223674\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "First apply the Rational Root Theorem.  The possible rational roots of:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ p(x)\ =\ \alpha_0x^n\ +\ \alpha_1x^{n-1}\ +\ \cdots\ +\ \alpha_{n-1}x\ +\ \alpha_n\">\r<BR>\n" );
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document.write( "are any rational number of the form\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \pm\frac{q}{p}\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q\ \in\ \mathbb{Z}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q\"> is a factor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha_n\"> and  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p\ \in\ \mathbb{Z}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p\"> is a factor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha_0\"> \r<BR>\n" );
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document.write( "Your lead coefficient being 1 simplifies things a little...your possible rational roots are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \pm1,\,\pm3,\,\pm5,\,\pm15\">\r<BR>\n" );
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document.write( "Use Synthetic Division and the Remainder Theorem to determine if any of these 8 possibilities are actually roots of the given equation.\r<BR>\n" );
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document.write( "If you are unfamiliar with the process of synthetic division, check out Purple Math's explanation at http://www.purplemath.com/modules/synthdiv.htm.\r<BR>\n" );
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document.write( "The Remainder Theorem says that the remainder when you use synthetic division with a divisor of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> is equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(a)\">, and if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(a)\ =\ 0\">, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\"> must be a root of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f\">\r<BR>\n" );
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document.write( "Fortunately for this particular problem, one of the rational roots actually works.  When you find the correct synthetic divisor, you will be left with the coefficients of a quadratic equation that can be solved with the quadratic formula to yield a conjugate pair of complex roots.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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