document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=972841>Question 972841</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm not really sure if this is an appropriate place to put this, since this is a precalculus question. Anyways, if you can help me solve this problem, thanks.\r<BR>\n" );
document.write( "\n" );
document.write( "Factor using the binomial theorem\r<BR>\n" );
document.write( "\n" );
document.write( "X^12 + 4x^9(y^3) + 6x^6(y^6) + 4x^3(y^9) + y^12\r<BR>\n" );
document.write( "\n" );
document.write( "I hope that's readable. This was a question on my test, and I had no idea what to do. I got (x+(33/2y))^6, but I don't remember how I got it. Either way, my answer's wrong. Finals are coming up, and this question might be on it. Again, thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #595065 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=595065\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your first clue is to look at the coefficients of your given polynomial:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^{12}\ +\ 4x^9y^3\ +\ 6x^6y^6\ +\ 4x^3y^9\ +\ y^{12}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Which are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1,\,4,\,6,\,4,\,1\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And this looks exactly like either the Number 4 row of Pascal's Triangle, or the sequence\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ {4 \choose 0},\ {4 \choose 1},\ {4 \choose 2},\ {4 \choose 3},\ {4 \choose 4},\ \">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "one of which I will hope makes sense based on your particular understanding of the binomial theorem.  Given that, we know from the binomial theorem that\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\alpha\ +\ \beta\right)^4\ =\ \alpha^4\ +\ 4\alpha^3\beta\ +\ 6\alpha^2\beta^2\ +\ 4\alpha\beta^3\ +\ \beta^4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's use a substitution to turn your original mud fence into something not quite so butt ugly.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\ =\ x^3\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \beta\ =\ y^3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now substitute.  Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^{12}\ =\ (x^3)^4\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^{9}\ =\ (x^3)^3\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^{6}\ =\ (x^3)^2\"> (and similarly for the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> terms), we can substitute <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \beta\"> into your original polynomial to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \alpha^4\ +\ 4\alpha^3\beta\ +\ 6\alpha^2\beta^2\ +\ 4\alpha\beta^3\ +\ \beta^4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "which we know from earlier to be equal to:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\alpha\ +\ \beta\right)^4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now just substitute back again to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(x^3\ +\ y^3\right)^4\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );