document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=338972>Question 338972</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation: \r<BR>\n" );
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document.write( "determine the second and fifth terms in the expanison of (2x-3y)^5? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #242977 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=242977\">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-3y%29%5E5 ALIGN=MIDDLE  ALT=\"%282x-3y%29%5E5\"   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" );
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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>\r<BR>\n" );
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document.write( "Looking at the row that starts with 1,5, etc, we can see that this row has the numbers:\r<BR>\n" );
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document.write( "1, 5, 10, 10, 5, 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-3y%29%5E5 ALIGN=MIDDLE  ALT=\"%282x-3y%29%5E5\"   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=-3y ALIGN=MIDDLE  ALT=\"-3y\"   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%28-3y%29%2Bhighlight%285%29%282x%29%28-3y%29%2Bhighlight%2810%29%282x%29%28-3y%29%2Bhighlight%2810%29%282x%29%28-3y%29%2Bhighlight%285%29%282x%29%28-3y%29%2Bhighlight%281%29%282x%29%28-3y%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%5E5%28-3y%29%5E0%2B%282x%29%28-3y%29%2B5%282x%29%28-3y%29%2B10%282x%29%28-3y%29%2B10%282x%29%28-3y%29%2B5%282x%29%28-3y%29%2B1%282x%29%28-3y%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%28-3y%29 ALIGN=MIDDLE  ALT=\"1%282x%29%28-3y%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=-3y ALIGN=MIDDLE  ALT=\"-3y\"   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%5E5%28-3y%29%5E0%2B%282x%29%5E4%28-3y%29%5E1%2B5%282x%29%28-3y%29%2B10%282x%29%28-3y%29%2B10%282x%29%28-3y%29%2B5%282x%29%28-3y%29%2B1%282x%29%28-3y%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=5%282x%29%28-3y%29 ALIGN=MIDDLE  ALT=\"5%282x%29%28-3y%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 4th power and raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3y ALIGN=MIDDLE  ALT=\"-3y\"   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=-3y ALIGN=MIDDLE  ALT=\"-3y\"   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%5E5%28-3y%29%5E0%2B5%282x%29%5E4%28-3y%29%5E1%2B10%282x%29%5E3%28-3y%29%5E2%2B10%282x%29%5E2%28-3y%29%5E3%2B5%282x%29%5E1%28-3y%29%5E4%2B1%282x%29%5E0%28-3y%29%5E5 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%2832x%5E5%29%28y%5E0%29%2B5%2816x%5E4%29%28-3y%5E1%29%2B10%288x%5E3%29%289y%5E2%29%2B10%284x%5E2%29%28-27y%5E3%29%2B5%282x%5E1%29%2881y%5E4%29%2B1%28x%5E0%29%28-243y%5E5%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%2832x%5E5%29%2B5%28-48x%5E4y%29%2B10%2872x%5E3y%5E2%29%2B10%28-108x%5E2y%5E3%29%2B5%28162xy%5E4%29%2B1%28-243y%5E5%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=32x%5E5-240x%5E4y%2B720x%5E3y%5E2-1080x%5E2y%5E3%2B810xy%5E4-243y%5E5 ALIGN=MIDDLE  ALT=\"32x%5E5-240x%5E4y%2B720x%5E3y%5E2-1080x%5E2y%5E3%2B810xy%5E4-243y%5E5\"   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-3y%29%5E5 ALIGN=MIDDLE  ALT=\"%282x-3y%29%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > expands and simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32x%5E5-240x%5E4y%2B720x%5E3y%5E2-1080x%5E2y%5E3%2B810xy%5E4-243y%5E5 ALIGN=MIDDLE  ALT=\"32x%5E5-240x%5E4y%2B720x%5E3y%5E2-1080x%5E2y%5E3%2B810xy%5E4-243y%5E5\"   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-3y%29%5E5=32x%5E5-240x%5E4y%2B720x%5E3y%5E2-1080x%5E2y%5E3%2B810xy%5E4-243y%5E5 ALIGN=MIDDLE  ALT=\"%282x-3y%29%5E5=32x%5E5-240x%5E4y%2B720x%5E3y%5E2-1080x%5E2y%5E3%2B810xy%5E4-243y%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From the last equation, we can see that the 2nd term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-240x%5E4y ALIGN=MIDDLE  ALT=\"-240x%5E4y\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the 5th term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=810xy%5E4 ALIGN=MIDDLE  ALT=\"810xy%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Shortcut: You can find the kth term of the expansion <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%5En ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > by using the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28nCk%29%2Aa%5E%28n-k%29%2Ab%5Ek ALIGN=MIDDLE  ALT=\"%28nCk%29%2Aa%5E%28n-k%29%2Ab%5Ek\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nCk ALIGN=MIDDLE  ALT=\"nCk\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the coefficient (either found with a calculator or by using Pascal's triangle).\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=I%20Need%20Algebra%20Help\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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document.write( "Also, feel free to check out my tutoring <a  rel=nofollow HREF=\"http://www.freewebs.com/jimthompson5910/home.html\">website</a>. \r<BR>\n" );
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document.write( "Jim </SPAN>\n" );
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