document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1154820>Question 1154820</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use Descartes rule of signs to find the possible positive, negative, and imaginary zeroes for:<BR>\n" );
document.write( "x^5 − 3x^4 + x^3 − 4x^2 + 5x − 1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #777337 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=777337\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E5+-3x%5E4+%2B+x%5E3+-4x%5E2+%2B+5x+-+1 ALIGN=MIDDLE  ALT=\"x%5E5+-3x%5E4+%2B+x%5E3+-4x%5E2+%2B+5x+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Here are the coefficients of our variable <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-4 ALIGN=MIDDLE  ALT=\"+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+5+ ALIGN=MIDDLE  ALT=\"+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.....    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "As can be seen, there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > changes.\r<BR>\n" );
document.write( "\n" );
document.write( "This means that there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+3 ALIGN=MIDDLE  ALT=\"+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=positive ALIGN=MIDDLE  ALT=\"positive\"   DISABLED_style= \"max-width:90%; height:auto;\" > real roots.\r<BR>\n" );
document.write( "\n" );
document.write( "To find the number of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=negative ALIGN=MIDDLE  ALT=\"negative\"   DISABLED_style= \"max-width:90%; height:auto;\" > real roots, substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the given polynomial:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E5+-3x%5E4+%2B+x%5E3+-4x%5E2+%2B+5x+-+1 ALIGN=MIDDLE  ALT=\"x%5E5+-3x%5E4+%2B+x%5E3+-4x%5E2+%2B+5x+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E5-3x%5E4-x%5E3-4x%5E2-5x-1 ALIGN=MIDDLE  ALT=\"-x%5E5-3x%5E4-x%5E3-4x%5E2-5x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "The coefficients are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As can be seen, there are<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0 ALIGN=MIDDLE  ALT=\"+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > changes. This means that there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=negative+ ALIGN=MIDDLE  ALT=\"negative+\"   DISABLED_style= \"max-width:90%; height:auto;\" >real roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E5+-3x%5E4+%2B+x%5E3+-4x%5E2+%2B+5x+-+1%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-5%2C+5%2C+-5%2C+5%2C+x%5E5+-3x%5E4+%2B+x%5E3+-4x%5E2+%2B+5x+-+1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "graph shows <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=positive ALIGN=MIDDLE  ALT=\"positive\"   DISABLED_style= \"max-width:90%; height:auto;\" > real roots, means there will be one <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pair ALIGN=MIDDLE  ALT=\"pair\"   DISABLED_style= \"max-width:90%; height:auto;\" > of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=imaginary ALIGN=MIDDLE  ALT=\"imaginary\"   DISABLED_style= \"max-width:90%; height:auto;\" >  roots too\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );