document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=231979>Question 231979</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have to find all roots of X^6 + 9X^3 + 8 = 0.<BR>\n" );
document.write( "I first let Y=X^3, then my equation became Y^2 + 9Y + 8 = 0. Solving for Y, I have Y = -8 OR -1.<BR>\n" );
document.write( "So I have two possible roots as X = -2 and X = -1. <BR>\n" );
document.write( "Since the highest power in the equation is 6, there are four other roots.<BR>\n" );
document.write( "How do you find the other roots?<BR>\n" );
document.write( "Thanks,<BR>\n" );
document.write( "Alan </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #171539 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=171539\">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=x%5E6+%2B+9x%5E3+%2B+8+=+0 ALIGN=MIDDLE  ALT=\"x%5E6+%2B+9x%5E3+%2B+8+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I took a direct approach and factored it into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3+%2B+8%29%28x%5E3+%2B+1%29+=+0 ALIGN=MIDDLE  ALT=\"%28x%5E3+%2B+8%29%28x%5E3+%2B+1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+=+-8 ALIGN=MIDDLE  ALT=\"x%5E3+=+-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2 ALIGN=MIDDLE  ALT=\"x+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I need the other 2 roots now. I proceed to cheat and<BR>\n" );
document.write( "apply what I know about complex roots, namely that <BR>\n" );
document.write( "they are equally spaced aound the complex plane. That means<BR>\n" );
document.write( "the 3 roots are 120 degrees apart. One complex root is<BR>\n" );
document.write( "30 degrees from the vertical in 1st quadrant, the other is<BR>\n" );
document.write( "30 degrees from vertical in 4th quadrant. The components of<BR>\n" );
document.write( "the complex roots are <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1+%2B+sqrt%283%29i ALIGN=MIDDLE  ALT=\"x+=+1+%2B+sqrt%283%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1st quadrant) and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1+-+sqrt%283%29i ALIGN=MIDDLE  ALT=\"x+=+1+-+sqrt%283%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" > (4th quadrant).<BR>\n" );
document.write( "I do the same for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+%2B+1 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+=+-1 ALIGN=MIDDLE  ALT=\"x%5E3+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The complex roots are <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F2+%2B+%28sqrt%283%29%2F2%29%2Ai ALIGN=MIDDLE  ALT=\"x+=+1%2F2+%2B+%28sqrt%283%29%2F2%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1st quadrant) and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1%2F2+-+%28sqrt%283%29%2F2%29%2Ai ALIGN=MIDDLE  ALT=\"x+=+1%2F2+-+%28sqrt%283%29%2F2%29%2Ai\"   DISABLED_style= \"max-width:90%; height:auto;\" > (4th quadrant<BR>\n" );
document.write( "I don't know an easy way to prove that's right, but<BR>\n" );
document.write( "I'm pretty sure it is\r<BR>\n" );
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document.write( " </SPAN>\n" );
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