document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=127623>Question 127623</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This problem is not from my textbook it is a worksheet problem.<BR>\n" );
document.write( "Use Descartes’s rule of signs to discuss the possibilities for the roots of the equation.<BR>\n" );
document.write( "4x3-9x2+2x+3=0<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 #93505 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=93505\">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=4x%5E3-9x%5E2%2B2x%2B3=0 ALIGN=MIDDLE  ALT=\"4x%5E3-9x%5E2%2B2x%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The signs are + - + +.  So there are 2 sign changes, 1st term to 2nd term, and 2nd term to 3rd term.\r<BR>\n" );
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document.write( "Two sign changes means that there are at most 2 positive roots.  There could also be 0 positive roots.\r<BR>\n" );
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document.write( "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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28-x%29%5E3-9%28-x%29%5E2%2B2%28-x%29%2B3=0 ALIGN=MIDDLE  ALT=\"4%28-x%29%5E3-9%28-x%29%5E2%2B2%28-x%29%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4x%5E3-9x%5E2-2x%2B3=0 ALIGN=MIDDLE  ALT=\"-4x%5E3-9x%5E2-2x%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The signs are - - - +.  So there is 1 sign change from the 3rd to the 4th term.\r<BR>\n" );
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document.write( "One sign change on <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;\" > means that there is exactly 1 negative real root.\r<BR>\n" );
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document.write( "So overall, this equation has either 2 positive real roots and 1 negative real root, or a conjugate pair of complex roots and 1 negative real root.\r<BR>\n" );
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document.write( "A graph of the function demonstrates that the first possibility is the actual case.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C4x%5E3-9x%5E2%2B2x%2B3%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-5%2C5%2C-5%2C5%2C4x%5E3-9x%5E2%2B2x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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