document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=8167>Question 8167</A>: <I><SPAN STYLE=\"word-break: break-all;\"> x^12 - 1<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 #4509 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=4509\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Ok,this gets a little messy, so please bear with me.\r<BR>\n" );
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document.write( "Factor: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E12+-+1 ALIGN=MIDDLE  ALT=\"x%5E12+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the difference of two squares.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E12+-+1+=+%28x%5E6+-+1%29%28x%5E6+%2B+1%29 ALIGN=MIDDLE  ALT=\"x%5E12+-+1+=+%28x%5E6+-+1%29%28x%5E6+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > NB: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E12+=+%28x%5E6%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E12+=+%28x%5E6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Take the first factor: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E6+-+1%29 ALIGN=MIDDLE  ALT=\"%28x%5E6+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and note that it is also the difference of two squares.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E6+-+1%29+=+%28x%5E3+-+1%29%28x%5E3+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x%5E6+-+1%29+=+%28x%5E3+-+1%29%28x%5E3+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice here that the 1st factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3+-+1%29 ALIGN=MIDDLE  ALT=\"%28x%5E3+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the difference of two cubes and that the second factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x%5E3+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the sum of two cubes.  These can be factored using the formulas:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%5E3+-+B%5E3%29+=+%28A+-+B%29%28A%5E2+%2B+AB+%2B+B%5E2%29 ALIGN=MIDDLE  ALT=\"%28A%5E3+-+B%5E3%29+=+%28A+-+B%29%28A%5E2+%2B+AB+%2B+B%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A%5E3+%2B+B%5E3%29+=+%28A+%2B+B%29%28A%5E2+-+AB+%2B+B%5E2%29 ALIGN=MIDDLE  ALT=\"%28A%5E3+%2B+B%5E3%29+=+%28A+%2B+B%29%28A%5E2+-+AB+%2B+B%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, we have: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3+-+1%29+=+%28x+-+1%29%28x%5E2+%2B+x+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x%5E3+-+1%29+=+%28x+-+1%29%28x%5E2+%2B+x+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3+%2B+1%29+=+%28x+%2B+1%29%28x%5E2+-+x+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x%5E3+%2B+1%29+=+%28x+%2B+1%29%28x%5E2+-+x+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, going back to the first line, let's look at the 2nd factor: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E6+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x%5E6+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and factor this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E6+%2B+1%29+=+%28x%5E2%2B+1%29%28x%5E4+-+x%5E2+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x%5E6+%2B+1%29+=+%28x%5E2%2B+1%29%28x%5E4+-+x%5E2+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now we can put it all together:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E12+-+1%29+=+%28x+-+1%29%28x+%2B+1%29%28x%5E2+%2B+1%29%28x%5E2+%2B+x+%2B+1%29%28x%5E2+-+x+%2B+1%29%28x%5E4+-+x%5E2+%2B+1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Whew!\r<BR>\n" );
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document.write( "You can verify this by multiplying all of these factors to see that you get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E12+-+1%29 ALIGN=MIDDLE  ALT=\"x%5E12+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > I'll leave this as an exercise for the student.\r<BR>\n" );
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document.write( "A final note:\r<BR>\n" );
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document.write( "I have left the factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x%5E2+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in its unfactored form, not because it can't be factored, but because I don't know whether or not you have covered complex numbers yet. But, in case you have, here's how you would factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+1 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+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=%28x%5E2+%2B+1%29+=+%28x+-+i%29%28x+%2B+i%29 ALIGN=MIDDLE  ALT=\"%28x%5E2+%2B+1%29+=+%28x+-+i%29%28x+%2B+i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i+=+sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"i+=+sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Using FOIL to check:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+i%29%28x+%2B+i%29+=+%28x%5E2+%2B+xi+-+xi+-+i%5E2%29 ALIGN=MIDDLE  ALT=\"%28x+-+i%29%28x+%2B+i%29+=+%28x%5E2+%2B+xi+-+xi+-+i%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > NB: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2+=+%28sqrt%28-1%29%29%5E2 ALIGN=MIDDLE  ALT=\"i%5E2+=+%28sqrt%28-1%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+i%29%28x+%2B+i%29+=+x%5E2+-+%28-1%29 ALIGN=MIDDLE  ALT=\"%28x+-+i%29%28x+%2B+i%29+=+x%5E2+-+%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+1%29 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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