document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1124403>Question 1124403</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many complex roots does the polynomial equation have?\r<BR>\n" );
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document.write( "5x^3&#8722;4x+1=0\r<BR>\n" );
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document.write( "2\r<BR>\n" );
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document.write( "3\r<BR>\n" );
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document.write( "4\r<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 #740720 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</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=740720\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "a polynomial equation of degree <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=exactly ALIGN=MIDDLE  ALT=\"exactly\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > roots\r<BR>\n" );
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document.write( "you have a polynomial equation of degree <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=exactly ALIGN=MIDDLE  ALT=\"exactly\"   DISABLED_style= \"max-width:90%; height:auto;\" > <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;\" > roots\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3-4x%2B1=0+ ALIGN=MIDDLE  ALT=\"5x%5E3-4x%2B1=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "here are they:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3-4x%2B1=0+ ALIGN=MIDDLE  ALT=\"5x%5E3-4x%2B1=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >...factor; add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2-5x%5E2 ALIGN=MIDDLE  ALT=\"5x%5E2-5x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4x ALIGN=MIDDLE  ALT=\"-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-5x ALIGN=MIDDLE  ALT=\"x-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=5x%5E3%2B5x%5E2-5x%5E2%26%238722%3B5x%2Bx%2B1=0+ ALIGN=MIDDLE  ALT=\"5x%5E3%2B5x%5E2-5x%5E2%26%238722%3B5x%2Bx%2B1=0+\"   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=%285x%5E3%2B5x%5E2%29-%285x%5E2%2B5x%29%2B%28x%2B1%29=0+ ALIGN=MIDDLE  ALT=\"%285x%5E3%2B5x%5E2%29-%285x%5E2%2B5x%29%2B%28x%2B1%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=5x%5E2%28x%2B1%29-5x%28x%2B1%29%2B%28x%2B1%29=0+ ALIGN=MIDDLE  ALT=\"5x%5E2%28x%2B1%29-5x%28x%2B1%29%2B%28x%2B1%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=%28x+%2B+1%29+%285x%5E2+-+5x+%2B+1%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+%2B+1%29+%285x%5E2+-+5x+%2B+1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "one real solution is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+1%29++=+0 ALIGN=MIDDLE  ALT=\"%28x+%2B+1%29++=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and, since your function is 3rd degree, there will be two more solutions  \r<BR>\n" );
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document.write( " use quadratic formula to solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%5E2+-+5x+%2B+1%29+=+0 ALIGN=MIDDLE  ALT=\"%285x%5E2+-+5x+%2B+1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "In this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=5 ALIGN=MIDDLE  ALT=\"a=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-5 ALIGN=MIDDLE  ALT=\"b=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=1 ALIGN=MIDDLE  ALT=\"c=1\"   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=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   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=x+=+%28-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%2A5%2A1+%29%29%2F%282%2A5%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-%28-5%29+%2B-+sqrt%28+%28-5%29%5E2-4%2A5%2A1+%29%29%2F%282%2A5%29+\"   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=x+=+%285+%2B-+sqrt%28+25-20+%29%29%2F10+ ALIGN=MIDDLE  ALT=\"x+=+%285+%2B-+sqrt%28+25-20+%29%29%2F10+\"   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=x+=+%285+%2B-+sqrt%28+5+%29%29%2F10+ ALIGN=MIDDLE  ALT=\"x+=+%285+%2B-+sqrt%28+5+%29%29%2F10+\"   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=x+=+%285%2F10+%2B-+sqrt%28+5+%29%2F10%29+ ALIGN=MIDDLE  ALT=\"x+=+%285%2F10+%2B-+sqrt%28+5+%29%2F10%29+\"   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=x+=+%281%2F2+%2B-+sqrt%28+5+%29%2F10%29+ ALIGN=MIDDLE  ALT=\"x+=+%281%2F2+%2B-+sqrt%28+5+%29%2F10%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "solutions:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F2+%2B+sqrt%28+5+%29%2F10+ ALIGN=MIDDLE  ALT=\"x+=+1%2F2+%2B+sqrt%28+5+%29%2F10+\"   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=x+=+1%2F2+-+sqrt%28+5+%29%2F10+ ALIGN=MIDDLE  ALT=\"x+=+1%2F2+-+sqrt%28+5+%29%2F10+\"   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+-5%2C+5%2C+-5%2C+5%2C+5x%5E3-4x%2B1%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-5%2C+5%2C+-5%2C+5%2C+5x%5E3-4x%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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