document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=351646>Question 351646</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2y^6 - 128 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #251275 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=251275\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Presumeably you want a complete factorization of this 6th degree polynomial expression.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large u\ =\ y^3\">.\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 \ \ \ \ \ \ \ \ \ \ 2y^6\ -\ 128\ =\ 2u^2\ -\ 128\">\r<BR>\n" );
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document.write( "Take out a 2:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\left(u^2\ -\ 64\right)\">\r<BR>\n" );
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document.write( "Factor the quadratic binomial:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\left(u\ -\ 8\right)\left(u\ +\ 8\right)\">\r<BR>\n" );
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document.write( "Replace the substitution:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\left(y^3\ -\ 8\right)\left(y^3\ +\ 8\right)\">\r<BR>\n" );
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document.write( "Use the sum and difference of cubes factorization:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\left(y\ -\ 2\right)\left(y^2\ +\ 2y\ +\ 4\right)\left(y\ +\ 2\right)\left(y^2\ -\ 2y\ +\ 4\right)\">\r<BR>\n" );
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document.write( "If you are factoring over <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \mathbb{R}\">, then you are done.  If you need to factor over <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \mathbb{C}\">, then continue using the quadratic formula to determine the four complex factors (verification left as an exercise for the student):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2\left(y\ -\ 2\right)\left(y\ +\ 1\ -\ i\sqrt{3}\right)\left(y\ +\ 1\ +\ i\sqrt{3}\right)\left(y\ +\ 2\right)\left(y\ -\ 1\ -\ i\sqrt{3}\right)\left(y\ -\ 1\ +\ i\sqrt{3}\right)\">\r<BR>\n" );
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document.write( "Ending with 6 binomial factors as the Fundamental Theorem of Algebra guarantees for a 6th degree polynomial.\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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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