document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=204277>Question 204277</A>: <I><SPAN STYLE=\"word-break: break-all;\"> i and -2i are two zeros of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29=x%5E6-2x%5E5%2B2x%5E4-10x%5E3-11x%5E2-8x-12 ALIGN=MIDDLE  ALT=\"p%28x%29=x%5E6-2x%5E5%2B2x%5E4-10x%5E3-11x%5E2-8x-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Find the other zeros and write p(x)as a product of linear factors. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #154231 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=154231\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Complex roots always come in conjugate pairs, so if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large i\"> is a root then so is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -i\">.  Likewise, if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -2i\"> is a root, then so is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2i\">.  Therefore, four of the factors you are looking for are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \ (x - i)(x + i)(x - 2i)(x + 2i)\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \ (x^2 + 1)(x^2 + 4)\">\r<BR>\n" );
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document.write( "The next step is to do two iterations of polynomial long division using each of the two 2nd degree factors above.  Polynomial long division is too difficult to render on this site -- at least I haven't figured out a way to actually show it in under 45 minutes to an hour of work.  So I'm just going to give you the result and if you need to see those steps, you can either send a thank-you note reply with your request, or you can go to:\r<BR>\n" );
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document.write( "www.purplemath.com/modules/polydiv2.htm\r<BR>\n" );
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document.write( "Where you can see an excellent explanation of the process\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \ \frac{x^6-2x^5+2x^4-10x^3-11x^2-8x-12}{x^2+1} = x^4-2x^3+x^2-8x-12\">\r<BR>\n" );
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document.write( "Then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \ \frac{x^4-2x^3+x^2-8x-12}{x^2+4}=x^2-2x-3\">\r<BR>\n" );
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document.write( "Which is then easily factored to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \ x^2 - 2x - 3 = (x - 3)(x + 1)\">\r<BR>\n" );
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document.write( "Therefore your six linear factors are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \ (x - i)(x + i)(x - 2i)(x + 2i)(x - 3)(x + 1)\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large e^{i\pi} + 1 = 0\"><BR>\n" );
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