document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163519>Question 1163519</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question:  I always thought, till now, that the number of possible roots for a polynomial is equal(or atmost) to its highest degreee. Also I have been counting each radical and complex root as two since they all have +/- values (just like +/- n will be counted as two). Then I came across the following polynomials (degree four and five):<BR>\n" );
document.write( "     4X^4 +8X^3 -5X^2 -2X +1 = 0<BR>\n" );
document.write( "     3X^5 +2X^4 -15X^3 -10X^2 +12X +8 = 0 <BR>\n" );
document.write( "The second one as expected has 5 roots: +/-1,  +/-2, -2/3<BR>\n" );
document.write( "But the first one which (degree-4) has FIVE roots! viz., +/-1/2, +/-√2 and -1.\r<BR>\n" );
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document.write( "So which of assumption is wrong?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #787632 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52803)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=787632\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "It is a &nbsp;<U>reminiscence</U> &nbsp;of the &nbsp;<U>Rational root theorem</U> &nbsp;in your mind.\r<BR>\n" );
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document.write( "The  &nbsp;<U>Rational root theorem</U>&nbsp;  says that all the rational roots of a polynomial with integer coefficients\r<BR>\n" );
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document.write( "are among the fractions &nbsp;&nbsp;+/- <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Fq ALIGN=MIDDLE  ALT=\"p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  &nbsp;where &nbsp;q &nbsp;is a divisor of the leading coefficient and &nbsp;p &nbsp;is a divisor of the constant term.\r<BR>\n" );
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document.write( "But this theorem &nbsp;DOES &nbsp;NOT &nbsp;state  \r<BR>\n" );
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document.write( "    a)  NEITHER  that all the roots of a polynomial are among these fractions\r\n" );
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document.write( "                 (irrational possible roots are DEFINITELY out of this set; complex roots ALSO are out of this set)\r\n" );
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document.write( "    b)  NOR      that each root goes two times with the \" + \"  and  \" - \" sign.\r\n" );
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document.write( "About this theorem, read this Wikipedia article \r<BR>\n" );
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document.write( "https://en.wikipedia.org/wiki/Rational_root_theorem\r<BR>\n" );
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document.write( "and have fan (!)\r<BR>\n" );
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