document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92194>Question 92194</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0395977258/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> According it Descartes' Rule of Signs, (a) how many positive real roots does each of the following have? (b) how many negative roots? i have no idea how to do this im trying to learn this stuff on my own and its hard please help\r<BR>\n" );
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document.write( " f (a) = a^5 - 4a^2 - 7    (a)__________   (b)_________\r<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 #67027 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=67027\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Using <a  rel=nofollow HREF=http://www.purplemath.com/modules/drofsign.htm>Descartes' Rule of Signs</a>, we can find the possible number of positive roots (x-intercepts that are positive) and negative roots (x-intercepts that are negative)\r<BR>\n" );
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document.write( "First lets find the number of possible positive real roots:\r<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E5-4a%5E2-7 ALIGN=MIDDLE  ALT=\"a%5E5-4a%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, simply count the sign changes\r<BR>\n" );
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document.write( "Here is the list of sign changes:\r<BR>\n" );
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document.write( "<ol><li><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;\" > (positive to negative)</li></ol>\r<BR>\n" );
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document.write( " (note: the rest of the terms have the same sign, so no extra sign changes occur) \r<BR>\n" );
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document.write( "So there is 1 sign change, this means there is a maximum of 1 positive root\r<BR>\n" );
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document.write( "So there is exactly one positive root\r<BR>\n" );
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document.write( "Now lets find the number of possible negative real roots\r<BR>\n" );
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document.write( "First we need to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-a%29 ALIGN=MIDDLE  ALT=\"f%28-a%29\"   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=%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;\" > Plug in -a (just replace every \"a\" with \"-a\")\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: if the exponent of the given term is odd, simply negate the sign of the term. If the term has an even exponent, then the sign of the term stays the same)\r<BR>\n" );
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document.write( "So <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( "------------------------\r<BR>\n" );
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document.write( "Now lets count the sign changes for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a%5E5-4a%5E2-7 ALIGN=MIDDLE  ALT=\"-a%5E5-4a%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "By looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-a%5E5-4a%5E2-7 ALIGN=MIDDLE  ALT=\"-a%5E5-4a%5E2-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that there are no sign changes (all the terms are negative).\r<BR>\n" );
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document.write( "So there are no negative roots\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "So there is 1 positive root and  0 negative roots </SPAN>\n" );
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