document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182855>Question 1182855</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The nature of the roots of the equation 3x^4 + 4x^3 + x - 1 = 0 is _____.\r<BR>\n" );
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document.write( "A. Three positive real roots and one negative real root.<BR>\n" );
document.write( "B. Three negative real roots and one positive real root.<BR>\n" );
document.write( "C. One negative real root and three complex roots.<BR>\n" );
document.write( "D. One positive real root, one negative real root and two complex roots.\r<BR>\n" );
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document.write( "--> Please help me on answering the correct nature of the roots on the equation and correct steps of doing it. Thank you so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #812948 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=812948\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "The nature of the roots of the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E4+%2B+4x%5E3+%2B+x+-+1+=+0+ ALIGN=MIDDLE  ALT=\"3x%5E4+%2B+4x%5E3+%2B+x+-+1+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "observe sign changes\r<BR>\n" );
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document.write( "signs are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E4+%2B+4x%5E3+%2B+x+-+1+ ALIGN=MIDDLE  ALT=\"3x%5E4+%2B+4x%5E3+%2B+x+-+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "+       +         +    -\r<BR>\n" );
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document.write( "=>  number of sign changes is <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( "it means => number of positive roots is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now 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=-x ALIGN=MIDDLE  ALT=\"-x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and observe sign changes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28-x%29%5E4+%2B+4%28-x%29%5E3+%2B+%28-x%29+-+1+ ALIGN=MIDDLE  ALT=\"3%28-x%29%5E4+%2B+4%28-x%29%5E3+%2B+%28-x%29+-+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=3x%5E4+-4x%5E3+-x+-+1 ALIGN=MIDDLE  ALT=\"3x%5E4+-4x%5E3+-x+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "+       -        -    -\r<BR>\n" );
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document.write( "=>  number of sign changes is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=> number of negative roots of original polynomial is<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+1 ALIGN=MIDDLE  ALT=\"+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so,<BR>\n" );
document.write( "=> number of positive roots is <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( "=> number of negative roots of original polynomial is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "=> number of imaginary roots of original polynomial is: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-%28p%2Bq%29 ALIGN=MIDDLE  ALT=\"n-%28p%2Bq%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .......where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is number of degree,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+p ALIGN=MIDDLE  ALT=\"+p\"   DISABLED_style= \"max-width:90%; height:auto;\" > is number of positive roots, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q ALIGN=MIDDLE  ALT=\"q\"   DISABLED_style= \"max-width:90%; height:auto;\" > is  number of negative roots\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=4 ALIGN=MIDDLE  ALT=\"n=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=1 ALIGN=MIDDLE  ALT=\"p=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q=1 ALIGN=MIDDLE  ALT=\"q=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=n-%28p%2Bq%29+=4-%281%2B1%29=2 ALIGN=MIDDLE  ALT=\"n-%28p%2Bq%29+=4-%281%2B1%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=> number of imaginary roots is <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( "then answer is\r<BR>\n" );
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document.write( "D. One positive real root, one negative real root and two complex roots\r<BR>\n" );
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document.write( " </SPAN>\n" );
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