document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=375956>Question 375956</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Expand <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x+-+y%29%5E6 ALIGN=MIDDLE  ALT=\"%282x+-+y%29%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I was able to expand to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%282x%5E6-y%5E0%29+%2B+6%282x%5E5-y%5E1%29+%2B+15%282x%5E4-y%5E2%29+%2B+20%282x%5E3-y%5E3%29+%2B+15%282x%5E2-y%5E4%29+%2B+6%282x%5E1-y%5E5%29+%2B+1%282x%5E0-y%5E6%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I multiplied the coefficients through and got:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E6+-+y%5E6+%2B+12x%5E5+-+6y%5E5+%2B+30x%5E4+-+15y%5E4+%2B+40x%5E3+-+20y%5E3+%2B+30x%5E2+-+15y%5E2+%2B+12x+-+6y ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Is this the complete expansion? \r<BR>\n" );
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document.write( "Thanks for your time! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #267376 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=267376\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-y%29%5E6 ALIGN=MIDDLE  ALT=\"%282x-y%29%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "To expand this, we're going to use binomial expansion. So let's look at Pascal's triangle:<BR>\n" );
document.write( "<center>1&nbsp; &nbsp;</center><BR>\n" );
document.write( "<center>1&nbsp; &nbsp;1&nbsp; &nbsp;</center><BR>\n" );
document.write( "<center>1&nbsp; &nbsp;2&nbsp; &nbsp;1&nbsp; &nbsp;</center><BR>\n" );
document.write( "<center>1&nbsp; &nbsp;3&nbsp; &nbsp;3&nbsp; &nbsp;1&nbsp; &nbsp;</center><BR>\n" );
document.write( "<center>1&nbsp; &nbsp;4&nbsp; &nbsp;6&nbsp; &nbsp;4&nbsp; &nbsp;1&nbsp; &nbsp;</center><BR>\n" );
document.write( "<center>1&nbsp; &nbsp;5&nbsp; &nbsp;10&nbsp; &nbsp;10&nbsp; &nbsp;5&nbsp; &nbsp;1&nbsp; &nbsp;</center><BR>\n" );
document.write( "<center>1&nbsp; &nbsp;6&nbsp; &nbsp;15&nbsp; &nbsp;20&nbsp; &nbsp;15&nbsp; &nbsp;6&nbsp; &nbsp;1&nbsp; &nbsp;</center>\r<BR>\n" );
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document.write( "Looking at the row that starts with 1,6, etc, we can see that this row has the numbers:\r<BR>\n" );
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document.write( "1, 6, 15, 20, 15, 6, and 1\r<BR>\n" );
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document.write( "These numbers will be the coefficients of our expansion. So to expand <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-y%29%5E6 ALIGN=MIDDLE  ALT=\"%282x-y%29%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, simply follow this procedure:\r<BR>\n" );
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document.write( "Write the first coefficient. Multiply that coefficient with the first binomial term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and then the second binomial term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y ALIGN=MIDDLE  ALT=\"-y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Repeat this until all of the coefficients have been written.\r<BR>\n" );
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document.write( "Once that has been done, add up the terms like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%29%282x%29%28-y%29%2Bhighlight%286%29%282x%29%28-y%29%2Bhighlight%2815%29%282x%29%28-y%29%2Bhighlight%2820%29%282x%29%28-y%29%2Bhighlight%2815%29%282x%29%28-y%29%2Bhighlight%286%29%282x%29%28-y%29%2Bhighlight%281%29%282x%29%28-y%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Notice how the coefficients are in front of each term.\r<BR>\n" );
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document.write( "However, we're not done yet.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%282x%29%5E6%28-y%29%5E0%2B%282x%29%28-y%29%2B6%282x%29%28-y%29%2B15%282x%29%28-y%29%2B20%282x%29%28-y%29%2B15%282x%29%28-y%29%2B6%282x%29%28-y%29%2B1%282x%29%28-y%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Looking at the first term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%282x%29%28-y%29 ALIGN=MIDDLE  ALT=\"1%282x%29%28-y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, raise  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 6th power and raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y ALIGN=MIDDLE  ALT=\"-y\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 0th power.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%282x%29%5E6%28-y%29%5E0%2B%282x%29%5E5%28-y%29%5E1%2B6%282x%29%28-y%29%2B15%282x%29%28-y%29%2B20%282x%29%28-y%29%2B15%282x%29%28-y%29%2B6%282x%29%28-y%29%2B1%282x%29%28-y%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Looking at the  second term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%282x%29%28-y%29 ALIGN=MIDDLE  ALT=\"6%282x%29%28-y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > raise  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 5th power and raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y ALIGN=MIDDLE  ALT=\"-y\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 1st power.\r<BR>\n" );
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document.write( "Continue this until you reach the final term.\r<BR>\n" );
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document.write( "Notice how the exponents of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > are stepping down and the exponents of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y ALIGN=MIDDLE  ALT=\"-y\"   DISABLED_style= \"max-width:90%; height:auto;\" >  are stepping up.\r<BR>\n" );
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document.write( "So the fully expanded expression should now look like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%282x%29%5E6%28-y%29%5E0%2B6%282x%29%5E5%28-y%29%5E1%2B15%282x%29%5E4%28-y%29%5E2%2B20%282x%29%5E3%28-y%29%5E3%2B15%282x%29%5E2%28-y%29%5E4%2B6%282x%29%5E1%28-y%29%5E5%2B1%282x%29%5E0%28-y%29%5E6 ALIGN=MIDDLE   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=1%2864x%5E6%29%28-y%5E0%29%2B6%2832x%5E5%29%28-y%29%5E1%2B15%2816x%5E4%29%28-y%29%5E2%2B20%288x%5E3%29%28-y%29%5E3%2B15%284x%5E2%29%28-y%29%5E4%2B6%282x%5E1%29%28-y%29%5E5%2B1%28x%5E0%29%28-y%29%5E6 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute the exponents\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2864x%5E6%29%2B6%28-32x%5E5y%29%2B15%2816x%5E4y%5E2%29%2B20%28-8x%5E3y%5E3%29%2B15%284x%5E2y%5E4%29%2B6%28-2xy%5E5%29%2B1%28y%5E6%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64x%5E6-192x%5E5y%2B240x%5E4y%5E2-160x%5E3y%5E3%2B60x%5E2y%5E4-12xy%5E5%2By%5E6 ALIGN=MIDDLE  ALT=\"64x%5E6-192x%5E5y%2B240x%5E4y%5E2-160x%5E3y%5E3%2B60x%5E2y%5E4-12xy%5E5%2By%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the terms with their coefficients\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-y%29%5E6 ALIGN=MIDDLE  ALT=\"%282x-y%29%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > expands and simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64x%5E6-192x%5E5y%2B240x%5E4y%5E2-160x%5E3y%5E3%2B60x%5E2y%5E4-12xy%5E5%2By%5E6 ALIGN=MIDDLE  ALT=\"64x%5E6-192x%5E5y%2B240x%5E4y%5E2-160x%5E3y%5E3%2B60x%5E2y%5E4-12xy%5E5%2By%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x-y%29%5E6=64x%5E6-192x%5E5y%2B240x%5E4y%5E2-160x%5E3y%5E3%2B60x%5E2y%5E4-12xy%5E5%2By%5E6 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If you need more help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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document.write( "Also, feel free to check out my <a  rel=nofollow HREF=\"http://www.freewebs.com/jimthompson5910/home.html\">tutoring website</a>\r<BR>\n" );
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document.write( "Jim </SPAN>\n" );
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