document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=693647>Question 693647</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2x^3-13x^2+24x-9=0 \r<BR>\n" );
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document.write( "How do I find zeros? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #427517 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=427517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Any rational zeros would be of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Fq ALIGN=MIDDLE  ALT=\"p%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > being a factor of the constant (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >),<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" > being a factor of the leading coefficient (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "Factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > are 1, 3, and 9.<BR>\n" );
document.write( "Factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > are 1, and 2.<BR>\n" );
document.write( "Possible rational zeros are -1/2, -3/2, -9/2, -1, -3, -9, 1/2, 3/2, 9/2, 1, 3, and 9.<BR>\n" );
document.write( "Descartes rule of signs helps narrow down your choices.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=2x%5E3-13x%5E2%2B24x-9 ALIGN=MIDDLE  ALT=\"P%28x%29=2x%5E3-13x%5E2%2B24x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > has 3 changes of sign going from one coefficient to the next. That means that it has 3 or 1 positive zeros. (It could be two different zeros if one is a double zero).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28-x%29=-2x%5E3-13x%5E2-24x-9 ALIGN=MIDDLE  ALT=\"P%28-x%29=-2x%5E3-13x%5E2-24x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > has 0 changes of sign, meaning that there are 0 negative zeros.<BR>\n" );
document.write( "Now we say that possible rational zeros are 1/2, 3/2, 9/2, 1, 3, and 9.<BR>\n" );
document.write( "Trying the possible zeros, using synthetic division, we find out that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3-13x%5E2%2B24x-9=%282x-1%29%28x%5E2-6x%2B9%29 ALIGN=MIDDLE  ALT=\"2x%5E3-13x%5E2%2B24x-9=%282x-1%29%28x%5E2-6x%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then we recognize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-6x%2B9=%28x-3%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-6x%2B9=%28x-3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and have the complete factorization,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3-13x%5E2%2B24x-9=%282x-1%29%28x-9%29%5E2 ALIGN=MIDDLE  ALT=\"2x%5E3-13x%5E2%2B24x-9=%282x-1%29%28x-9%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "which tells us that the zeros are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283%29 ALIGN=MIDDLE  ALT=\"highlight%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with multiplicity 2, a double zero), and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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