document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=166504>Question 166504</A>: <I><SPAN STYLE=\"word-break: break-all;\"> These will be the last questions.  Please help<BR>\n" );
document.write( "1.  List all of the possible rational zeros of f(x)=3x^5-7x^3+2x-15.<BR>\n" );
document.write( "2.  State the number of positive real zeros, negative real zeros, and imaginary zeros for g(x)=9x^3-7x^2+10x-4.<BR>\n" );
document.write( "I must show my work.  No textbook.  Have been absent from school and have a big test on Monday.  Need to know how to do these for the test.<BR>\n" );
document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #122706 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=122706\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 1\r<BR>\n" );
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document.write( "Any rational zero can be found through this equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Roots=\frac{p}{q}\"> where p and q are the factors of the last and first coefficients\r<BR>\n" );
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document.write( "So let's list the factors of -15 (the last coefficient):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p=\pm1, \pm3, \pm5, \pm15\">\r<BR>\n" );
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document.write( "Now let's list the factors of 3 (the first coefficient):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q=\pm1, \pm3\">\r<BR>\n" );
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document.write( "Now let's divide each factor of the last coefficient by each factor of the first coefficient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{1}, \frac{1}{3}, \frac{3}{1}, \frac{3}{3}, \frac{5}{1}, \frac{5}{3}, \frac{15}{1}, \frac{15}{3}, \frac{-1}{1}, \frac{-1}{3}, \frac{-3}{1}, \frac{-3}{3}, \frac{-5}{1}, \frac{-5}{3}, \frac{-15}{1}, \frac{-15}{3}\">\r<BR>\n" );
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document.write( "Now simplify\r<BR>\n" );
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document.write( "These are all the distinct rational zeros of the function that could occur\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  1, \frac{1}{3}, 3, 5, \frac{5}{3}, 15, -1, \frac{-1}{3}, -3, -5, \frac{-5}{3}, -15\">\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "Now let's use Descartes Rule of Signs to find the number of positive, negative,and imaginary zeros.\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=9x%5E3-7x%5E2%2B10x-4 ALIGN=MIDDLE  ALT=\"f%28x%29=9x%5E3-7x%5E2%2B10x-4\"   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=9x%5E3 ALIGN=MIDDLE  ALT=\"9x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7x%5E2 ALIGN=MIDDLE  ALT=\"-7x%5E2\"   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( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7x%5E2 ALIGN=MIDDLE  ALT=\"-7x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x ALIGN=MIDDLE  ALT=\"10x\"   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=10x ALIGN=MIDDLE  ALT=\"10x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   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 3 sign changes for the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=9x%5E3-7x%5E2%2B10x-4 ALIGN=MIDDLE  ALT=\"f%28x%29=9x%5E3-7x%5E2%2B10x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So there are 3 or 1 positive zeros\r<BR>\n" );
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document.write( "So if there are 3 positive zeros, then there will be NO imaginary zeros (since there a total of 3 zeros)\r<BR>\n" );
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document.write( "Or...\r<BR>\n" );
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document.write( "If there is only 1 positive zero, then there will be 2 imaginary 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=9%28-x%29%5E3-7%28-x%29%5E2%2B10%28-x%29-4 ALIGN=MIDDLE  ALT=\"f%28-x%29=9%28-x%29%5E3-7%28-x%29%5E2%2B10%28-x%29-4\"   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=-9x%5E3-7x%5E2-10x-4 ALIGN=MIDDLE  ALT=\"f%28-x%29=-9x%5E3-7x%5E2-10x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\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=-9x%5E3-7x%5E2-10x-4 ALIGN=MIDDLE  ALT=\"f%28-x%29=-9x%5E3-7x%5E2-10x-4\"   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=-9x%5E3 ALIGN=MIDDLE  ALT=\"-9x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7x%5E2 ALIGN=MIDDLE  ALT=\"-7x%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=-7x%5E2 ALIGN=MIDDLE  ALT=\"-7x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10x ALIGN=MIDDLE  ALT=\"-10x\"   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=-10x ALIGN=MIDDLE  ALT=\"-10x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   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 expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-x%29=-9x%5E3-7x%5E2-10x-4 ALIGN=MIDDLE  ALT=\"f%28-x%29=-9x%5E3-7x%5E2-10x-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So there are 0 negative zeros\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So here are the possibilities:\r<BR>\n" );
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document.write( "3 positive zeros, 0 negative zeros, and 0 imaginary zeros\r<BR>\n" );
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document.write( "Or...\r<BR>\n" );
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document.write( "1 positive zero, 0 negative zeros, and 2 imaginary zeros </SPAN>\n" );
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