document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=976976>Question 976976</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hey ! I need help solving this.\r<BR>\n" );
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document.write( "f(x) = x^4 &#8722; x^3 &#8722; 20x^2 \r<BR>\n" );
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document.write( "What are all the zeros of the function and determine the multiplicity of each zero and the number of turning points of the graph of the function. \r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #598463 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</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=598463\">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=f%28x%29+=+x%5E4+-x%5E3+-+20x%5E2+ ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%5E4+-x%5E3+-+20x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "set <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=0 ALIGN=MIDDLE  ALT=\"f%28x%29+=0\"   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=0=+x%5E4+-x%5E3+-+20x%5E2+ ALIGN=MIDDLE  ALT=\"0=+x%5E4+-x%5E3+-+20x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >....factor out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   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=0=+%28x%5E2+-+x+-+20%29x%5E2+ ALIGN=MIDDLE  ALT=\"0=+%28x%5E2+-+x+-+20%29x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >....factor completely <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2+-+x+-+20%29+ ALIGN=MIDDLE  ALT=\"%28x%5E2+-+x+-+20%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <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=4x-5x ALIGN=MIDDLE  ALT=\"4x-5x\"   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=0=+%28x%5E2+-+4x%2B5x+-+20%29x%5E2+ ALIGN=MIDDLE  ALT=\"0=+%28x%5E2+-+4x%2B5x+-+20%29x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >......group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=+%28%28x%5E2+-+4x%29%2B%285x+-+20%29%29x%5E2+ ALIGN=MIDDLE  ALT=\"0=+%28%28x%5E2+-+4x%29%2B%285x+-+20%29%29x%5E2+\"   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=0=+%28x%28x+-+4%29%2B5%28x+-+4%29%29x%5E2+ ALIGN=MIDDLE  ALT=\"0=+%28x%28x+-+4%29%2B5%28x+-+4%29%29x%5E2+\"   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=0=+%28x%2B4%29%28x-5%29x%5E2+ ALIGN=MIDDLE  ALT=\"0=+%28x%2B4%29%28x-5%29x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, zeros are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=+%28x%2B4%29+ ALIGN=MIDDLE  ALT=\"0=+%28x%2B4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >-the multiplicity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=+%28x-5%29+ ALIGN=MIDDLE  ALT=\"0=+%28x-5%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >-the multiplicity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=+x%5E2+ ALIGN=MIDDLE  ALT=\"0=+x%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><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;\" >, -the multiplicity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   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=+graph%28+600%2C+600%2C+-20%2C+20%2C+-150%2C+90%2C++%28x%2B4%29%28x-5%29x%5E2%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-20%2C+20%2C+-150%2C+90%2C++%28x%2B4%29%28x-5%29x%5E2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "as you can see, there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > turning points of the graph of the function\r<BR>\n" );
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document.write( " </SPAN>\n" );
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