document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1010745>Question 1010745</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Possible number of imaginary zeros in g(X)=x^4+3x^3+7x^2-6x-13 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #626213 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</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=626213\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The analysis you are using is called Descartes' rule of signs.\r<BR>\n" );
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document.write( "When you have determined the number of sign changes for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then you know the (maximum) possible number of positive zeros for your function.\r<BR>\n" );
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document.write( "You then have to evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get the (maximum) possible 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;\" > zeros.\r<BR>\n" );
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document.write( "Positive and negative possible zeros always decrease by two's. You can't go below none.\r<BR>\n" );
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document.write( "In your problem:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=x%5E4%2B3x%5E3%2B7x%5E2-6x-13+ ALIGN=MIDDLE  ALT=\"g%28x%29=x%5E4%2B3x%5E3%2B7x%5E2-6x-13+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "there 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;\" > sign changes for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+%2B+x%29 ALIGN=MIDDLE  ALT=\"%28+%2B+x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which means there is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  positive real zero\r<BR>\n" );
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document.write( "now, rewrite the given polynomial by substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <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" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=%28-x%29%5E4%2B3%28-x%29%5E3%2B7%28-x%29%5E2-6%28-x%29-13+ ALIGN=MIDDLE  ALT=\"g%28x%29=%28-x%29%5E4%2B3%28-x%29%5E3%2B7%28-x%29%5E2-6%28-x%29-13+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=x%5E4+-3x%5E3%2B7x%5E2%2B6x-13+ ALIGN=MIDDLE  ALT=\"g%28x%29=x%5E4+-3x%5E3%2B7x%5E2%2B6x-13+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > sign changes for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+-x%29 ALIGN=MIDDLE  ALT=\"%28+-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which means  there are a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=maximum ALIGN=MIDDLE  ALT=\"maximum\"   DISABLED_style= \"max-width:90%; height:auto;\" > of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=two ALIGN=MIDDLE  ALT=\"two\"   DISABLED_style= \"max-width:90%; height:auto;\" > negative  real zeros\r<BR>\n" );
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document.write( "since we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4th ALIGN=MIDDLE  ALT=\"4th\"   DISABLED_style= \"max-width:90%; height:auto;\" > degree function,means there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > zeros in all, and we know that complex zeros <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=always ALIGN=MIDDLE  ALT=\"always\"   DISABLED_style= \"max-width:90%; height:auto;\" > come in pairs, it means there will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > negative zero and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > imaginary or complex zeros\r<BR>\n" );
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document.write( "so, in all will be:\r<BR>\n" );
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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;\" >  positive real zero\r<BR>\n" );
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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;\" >  negative real zero\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  complex zeros\r<BR>\n" );
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