document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=120295>Question 120295</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (3x+y)^5\r<BR>\n" );
document.write( "\n" );
document.write( "Set up is <BR>\n" );
document.write( " 1     5     10      10      5      1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #88168 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=88168\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2By%29%5E5 ALIGN=MIDDLE  ALT=\"%283x%2By%29%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Looking at the row that starts with 1,5, etc, we can see that this row has the numbers:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1, 5, 10, 10, 5, and 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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=%283x%2By%29%5E5 ALIGN=MIDDLE  ALT=\"%283x%2By%29%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, simply follow this procedure:\r<BR>\n" );
document.write( "\n" );
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=3x ALIGN=MIDDLE  ALT=\"3x\"   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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Once that has been done, add up every term like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%281%29%283x%29%28y%29%2Bhighlight%285%29%283x%29%28y%29%2Bhighlight%2810%29%283x%29%28y%29%2Bhighlight%2810%29%283x%29%28y%29%2Bhighlight%285%29%283x%29%28y%29%2Bhighlight%281%29%283x%29%28y%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Notice how the coefficients are in front of each term.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "However, we're not done yet.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%283x%29%5E5%28y%29%5E0%2B%283x%29%28y%29%2B10%283x%29%28y%29%2B10%283x%29%28y%29%2B5%283x%29%28y%29%2B1%283x%29%28y%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%283x%29%28y%29 ALIGN=MIDDLE  ALT=\"1%283x%29%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, raise  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   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 0th power.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%283x%29%5E5%28y%29%5E0%2B%283x%29%5E4%28y%29%5E1%2B10%283x%29%28y%29%2B10%283x%29%28y%29%2B5%283x%29%28y%29%2B1%283x%29%28y%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%283x%29%28y%29 ALIGN=MIDDLE  ALT=\"5%283x%29%28y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > raise  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   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=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 1st power.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Continue this until you reach the final term.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how the exponents of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   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" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the fully expanded expression should now look like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%283x%29%5E5%28y%29%5E0%2B5%283x%29%5E4%28y%29%5E1%2B10%283x%29%5E3%28y%29%5E2%2B10%283x%29%5E2%28y%29%5E3%2B5%283x%29%5E1%28y%29%5E4%2B1%283x%29%5E0%28y%29%5E5 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%28243x%5E5%29%28y%5E0%29%2B5%2881x%5E4%29%28y%5E1%29%2B10%2827x%5E3%29%28y%5E2%29%2B10%289x%5E2%29%28y%5E3%29%2B5%283x%5E1%29%28y%5E4%29%2B1%28x%5E0%29%28y%5E5%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute the exponents\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%28243x%5E5%29%2B5%2881x%5E4y%29%2B10%2827x%5E3y%5E2%29%2B10%289x%5E2y%5E3%29%2B5%283xy%5E4%29%2B1%28y%5E5%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=243x%5E5%2B405x%5E4y%2B270x%5E3y%5E2%2B90x%5E2y%5E3%2B15xy%5E4%2By%5E5 ALIGN=MIDDLE  ALT=\"243x%5E5%2B405x%5E4y%2B270x%5E3y%5E2%2B90x%5E2y%5E3%2B15xy%5E4%2By%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the terms with their coefficients\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2By%29%5E5 ALIGN=MIDDLE  ALT=\"%283x%2By%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=243x%5E5%2B405x%5E4y%2B270x%5E3y%5E2%2B90x%5E2y%5E3%2B15xy%5E4%2By%5E5 ALIGN=MIDDLE  ALT=\"243x%5E5%2B405x%5E4y%2B270x%5E3y%5E2%2B90x%5E2y%5E3%2B15xy%5E4%2By%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2By%29%5E5=243x%5E5%2B405x%5E4y%2B270x%5E3y%5E2%2B90x%5E2y%5E3%2B15xy%5E4%2By%5E5 ALIGN=MIDDLE  ALT=\"%283x%2By%29%5E5=243x%5E5%2B405x%5E4y%2B270x%5E3y%5E2%2B90x%5E2y%5E3%2B15xy%5E4%2By%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );