document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=222493>Question 222493</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use the Binomial Theorem to expand (x - 2y)^3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #166626 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=166626\">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=%28x-2y%29%5E3 ALIGN=MIDDLE  ALT=\"%28x-2y%29%5E3\"   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>\r<BR>\n" );
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document.write( "Looking at the row that starts with 1,3, etc, we can see that this row has the numbers:\r<BR>\n" );
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document.write( "1, 3, 3, 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=%28x-2y%29%5E3 ALIGN=MIDDLE  ALT=\"%28x-2y%29%5E3\"   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=x ALIGN=MIDDLE  ALT=\"x\"   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=-2y ALIGN=MIDDLE  ALT=\"-2y\"   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%28x%29%28-2y%29%2Bhighlight%283%29%28x%29%28-2y%29%2Bhighlight%283%29%28x%29%28-2y%29%2Bhighlight%281%29%28x%29%28-2y%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%28x%29%5E3%28-2y%29%5E0%2B%28x%29%28-2y%29%2B3%28x%29%28-2y%29%2B3%28x%29%28-2y%29%2B1%28x%29%28-2y%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%28x%29%28-2y%29 ALIGN=MIDDLE  ALT=\"1%28x%29%28-2y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, raise  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 3rd power and raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2y ALIGN=MIDDLE  ALT=\"-2y\"   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%28x%29%5E3%28-2y%29%5E0%2B%28x%29%5E2%28-2y%29%5E1%2B3%28x%29%28-2y%29%2B3%28x%29%28-2y%29%2B1%28x%29%28-2y%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=3%28x%29%28-2y%29 ALIGN=MIDDLE  ALT=\"3%28x%29%28-2y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > raise  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 2nd power and raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2y ALIGN=MIDDLE  ALT=\"-2y\"   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=x ALIGN=MIDDLE  ALT=\"x\"   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=-2y ALIGN=MIDDLE  ALT=\"-2y\"   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%28x%29%5E3%28-2y%29%5E0%2B3%28x%29%5E2%28-2y%29%5E1%2B3%28x%29%5E1%28-2y%29%5E2%2B1%28x%29%5E0%28-2y%29%5E3 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%28x%5E3%29%28y%5E0%29%2B3%28x%5E2%29%28-2y%5E1%29%2B3%28x%5E1%29%284y%5E2%29%2B1%28x%5E0%29%28-8y%5E3%29 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%28x%5E3%29%2B3%28-2x%5E2y%29%2B3%284xy%5E2%29%2B1%28-8y%5E3%29 ALIGN=MIDDLE  ALT=\"1%28x%5E3%29%2B3%28-2x%5E2y%29%2B3%284xy%5E2%29%2B1%28-8y%5E3%29\"   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=x%5E3-6x%5E2y%2B12xy%5E2-8y%5E3 ALIGN=MIDDLE  ALT=\"x%5E3-6x%5E2y%2B12xy%5E2-8y%5E3\"   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=%28x-2y%29%5E3 ALIGN=MIDDLE  ALT=\"%28x-2y%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > expands and simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-6x%5E2y%2B12xy%5E2-8y%5E3 ALIGN=MIDDLE  ALT=\"x%5E3-6x%5E2y%2B12xy%5E2-8y%5E3\"   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=%28x-2y%29%5E3=x%5E3-6x%5E2y%2B12xy%5E2-8y%5E3 ALIGN=MIDDLE  ALT=\"%28x-2y%29%5E3=x%5E3-6x%5E2y%2B12xy%5E2-8y%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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