document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=204288>Question 204288</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is this right; according to Descartes' rule of signs how many negative roots, and how many positive real roots are there in this problem; f(x) =3x^3+9x^2+8x so there are 3 positive real roots and 1 negative root??  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #154235 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=154235\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Please show your work as to how you got your answers. I'm not sure how you got them and where you went wrong.\r<BR>\n" );
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document.write( "First count the sign changes of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=3x%5E3%2B9x%5E2%2B8x ALIGN=MIDDLE  ALT=\"f%28x%29=3x%5E3%2B9x%5E2%2B8x\"   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=3x%5E3 ALIGN=MIDDLE  ALT=\"3x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2 ALIGN=MIDDLE  ALT=\"9x%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=9x%5E2 ALIGN=MIDDLE  ALT=\"9x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x ALIGN=MIDDLE  ALT=\"8x\"   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 function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=3x%5E3%2B9x%5E2%2B8x ALIGN=MIDDLE  ALT=\"f%28x%29=3x%5E3%2B9x%5E2%2B8x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So there are 0 positive zeros\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29=3%28-x%29%5E3%2B9%28-x%29%5E2%2B8%28-x%29 ALIGN=MIDDLE  ALT=\"f%28-x%29=3%28-x%29%5E3%2B9%28-x%29%5E2%2B8%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now let's replace each <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29=-3x%5E3%2B9x%5E2-8x ALIGN=MIDDLE  ALT=\"f%28-x%29=-3x%5E3%2B9x%5E2-8x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify. Note: only the terms with odd exponents will have a change in sign.\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-x%29=-3x%5E3%2B9x%5E2-8x ALIGN=MIDDLE  ALT=\"f%28-x%29=-3x%5E3%2B9x%5E2-8x\"   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=-3x%5E3 ALIGN=MIDDLE  ALT=\"-3x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2 ALIGN=MIDDLE  ALT=\"9x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, there is a sign change from negative to positive \r<BR>\n" );
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document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2 ALIGN=MIDDLE  ALT=\"9x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8x ALIGN=MIDDLE  ALT=\"-8x\"   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( "So there are 2 sign changes for the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29=-3x%5E3%2B9x%5E2-8x ALIGN=MIDDLE  ALT=\"f%28-x%29=-3x%5E3%2B9x%5E2-8x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So there are 2 or 0 negative zeros. Note: you count down by 2 (since complex zeros come in conjugate pairs).\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%28x%29=3x%5E3%2B9x%5E2%2B8x+ ALIGN=MIDDLE  ALT=\"f%28x%29=3x%5E3%2B9x%5E2%2B8x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you'll see that there are no positive or negative real zeros. It turns out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only zero (which is neither positive nor negative). </SPAN>\n" );
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