document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=204290>Question 204290</A>: <I><SPAN STYLE=\"word-break: break-all;\"> could anyone out there help me out here: how many positive real roots and how many negative roots does the problem have; f(a)=a^5-4a^2-7 (using Descartes rule of signs?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #154234 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=154234\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "First count the sign changes of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29=a%5E5-4a%5E2-7 ALIGN=MIDDLE  ALT=\"f%28a%29=a%5E5-4a%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E5 ALIGN=MIDDLE  ALT=\"a%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4a%5E2 ALIGN=MIDDLE  ALT=\"-4a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is a sign change from positive to negative \r<BR>\n" );
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document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4a%5E2 ALIGN=MIDDLE  ALT=\"-4a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7 ALIGN=MIDDLE  ALT=\"-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is no change in sign\r<BR>\n" );
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document.write( "So there is 1 sign change for the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29=a%5E5-4a%5E2-7 ALIGN=MIDDLE  ALT=\"f%28a%29=a%5E5-4a%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So there is 1 positive real zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-a%29=%28-a%29%5E5-4%28-a%29%5E2-7 ALIGN=MIDDLE  ALT=\"f%28-a%29=%28-a%29%5E5-4%28-a%29%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now let's replace each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a ALIGN=MIDDLE  ALT=\"-a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-a%29=-a%5E5-4a%5E2-7 ALIGN=MIDDLE  ALT=\"f%28-a%29=-a%5E5-4a%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. Note: only the terms with odd exponents will have a sign change.\r<BR>\n" );
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document.write( "Now let's count the sign changes of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-a%29=-a%5E5-4a%5E2-7 ALIGN=MIDDLE  ALT=\"f%28-a%29=-a%5E5-4a%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a%5E5 ALIGN=MIDDLE  ALT=\"-a%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4a%5E2 ALIGN=MIDDLE  ALT=\"-4a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is no change in sign\r<BR>\n" );
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document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4a%5E2 ALIGN=MIDDLE  ALT=\"-4a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7 ALIGN=MIDDLE  ALT=\"-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is no change in sign\r<BR>\n" );
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document.write( "So there are no sign changes for the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-a%29=-a%5E5-4a%5E2-7 ALIGN=MIDDLE  ALT=\"f%28-a%29=-a%5E5-4a%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So there are 0 negative real zeros\r<BR>\n" );
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document.write( "Note: if you graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28a%29=a%5E5-4a%5E2-7 ALIGN=MIDDLE  ALT=\"f%28a%29=a%5E5-4a%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you will find that there is indeed one positive real zero. </SPAN>\n" );
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