document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=189028>Question 189028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given P(x)=6x^3+x^2+2x+3, how many POSSIBLE roots of P(x) are there? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #141795 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=141795\">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( "This is a 3rd degree polynomial function, hence the Fundamental Theorem of Algebra tells us that there are exactly 3 factors of the polynomial and therefore 3 roots -- no more and no less.  But I don't think that is the question you meant to ask.  The 3 roots are at least one real number which may be irrational and the other two may be either a conjugate pair of complex numbers or they may be real also but, again, no guarantee that they are rational.\r<BR>\n" );
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document.write( "The question you meant to ask, I think, is:\r<BR>\n" );
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document.write( "Given <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P(x)=6x^3+x^2+2x+3}\">, how many POSSIBLE <i><b>rational</b></i> roots of P(x) are there?\r<BR>\n" );
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document.write( "To answer that, we need the Rational Root Theorem.\r<BR>\n" );
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document.write( "First, find all of the integer factors of the constant term, namely 3 in your problem.  3 is prime so the only integer factors are 1 and 3.  Call this <i>p</i>.  So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large p\,\in\,\{1,3}\">\r<BR>\n" );
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document.write( "Next, find all of the integer factors of the lead coefficient, namely 6 in your problem.  The factors are 1, 2, 3, and 6.  Call this <i>q</i>.  So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large q\,\in\,\{1,2,3,6}\">\r<BR>\n" );
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document.write( "The list of possible rational roots of your polynomial function can then be derived by substituting all of the possible values for <i>p</i> and <i>q</i> into:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \pm \frac{p}{q}\">\r<BR>\n" );
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document.write( "So we have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \pm \frac{1}{1} = \pm 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \pm \frac{1}{2} \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \pm \frac{1}{3}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \pm \frac{1}{6}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \pm \frac{3}{1} = \pm 3\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \pm \frac{3}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \pm \frac{3}{3} = +\pm 1\"> (but you already have this in the list)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x = \pm \frac{3}{6} = \pm \frac{1}{2}\"> (you already have this in the list, too)\r<BR>\n" );
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document.write( "Remember, this is a list of possibilities.  No more than three but possibly none of these may actually be roots.  The only guarantee is that any rational number <i><b>not</b></i> in this list is <i><b>not</b></i> a root of this function.\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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