document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=61330>Question 61330</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I seem to be having a small problem understanding Binomial Theorems as far as substitution and Expansion.  Your assistancewith this problem would be greatly appreciated:\r<BR>\n" );
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document.write( "It states:  Use substitution to find the expansion of (c-2i)^5.  Simplify your final result.\r<BR>\n" );
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document.write( "Please help!<BR>\n" );
document.write( "Thanks in advance! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #42212 by </B><A HREF=/tutors/aboutme.mpl?userid=Cintchr><B>Cintchr(481)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Cintchr><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=42212\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28c-2i%29%5E5+ ALIGN=MIDDLE  ALT=\"+%28c-2i%29%5E5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "let's expand this out<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28c-2i%29%28c-2i%29%28c-2i%29%28c-2i%29%28c-2i%29+ ALIGN=MIDDLE  ALT=\"+%28c-2i%29%28c-2i%29%28c-2i%29%28c-2i%29%28c-2i%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Looking at just the first 2 quantities ... let's FOIL them<BR>\n" );
document.write( "F: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+%2A+c+=+c%5E2+ ALIGN=MIDDLE  ALT=\"+c+%2A+c+=+c%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "O: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c+%2A+-2i+=+-2ci+ ALIGN=MIDDLE  ALT=\"+c+%2A+-2i+=+-2ci+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-2i+%2A+c+=+-2ci+ ALIGN=MIDDLE  ALT=\"+-2i+%2A+c+=+-2ci+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "L: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-2i+%2A+-2i+=+4i%5E2+=4%28-1%29+=+-4+ ALIGN=MIDDLE  ALT=\"+-2i+%2A+-2i+=+4i%5E2+=4%28-1%29+=+-4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "remember i^2 = -1<BR>\n" );
document.write( "Combine like terms<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c%5E2+-4ci+-4+ ALIGN=MIDDLE  ALT=\"+c%5E2+-4ci+-4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now the 3rd and 4th terms when FOILed, give the same response.<BR>\n" );
document.write( "You now have the following<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28c%5E2+-4ci+-4%29%28c%5E2+-4ci+-4%29%28c-2i%29+ ALIGN=MIDDLE  ALT=\"+%28c%5E2+-4ci+-4%29%28c%5E2+-4ci+-4%29%28c-2i%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "now multiply out the two trinomials<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28c%5E2%2Ac%5E2%29%2B%28c%5E2%2A-4ci%29%2B%28c%5E2%2A-4%29%2B%28-4ci%2Ac%5E2%29%2B%28-4ci%2A-4ci%29%2B%28-4ci%2A-4%29%2B%28-4%2Ac%5E2%29%2B%28-4%2A-4ci%29%2B%28-4%2A-4%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "simplify<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c%5E4+-4c%5E3i+-4c%5E2+-4c%5E3i+-16c%5E2i%5E2+%2B16ci+-4c%5E2+%2B16ci+%2B16+ ALIGN=MIDDLE  ALT=\"+c%5E4+-4c%5E3i+-4c%5E2+-4c%5E3i+-16c%5E2i%5E2+%2B16ci+-4c%5E2+%2B16ci+%2B16+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "multiply, working the i^2 as well<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c%5E4+-8c%5E3i+-8c%5E2+-16c%5E2%28-1%29+%2B16ci+%2B16+ ALIGN=MIDDLE  ALT=\"+c%5E4+-8c%5E3i+-8c%5E2+-16c%5E2%28-1%29+%2B16ci+%2B16+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "combine like terms<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c%5E4+-+8c%5E3i+-8c%5E2+%2B16c%5E2+%2B16ci+%2B16+ ALIGN=MIDDLE  ALT=\"+c%5E4+-+8c%5E3i+-8c%5E2+%2B16c%5E2+%2B16ci+%2B16+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c%5E4+-+8c%5E3i+%2B8c%5E2+%2B16ci+%2B16+ ALIGN=MIDDLE  ALT=\"+c%5E4+-+8c%5E3i+%2B8c%5E2+%2B16ci+%2B16+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now using this quantity ... multiply it by the 5th (c-2i)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28c-2i%29%28c%5E4+-8c%5E3i+%2B8c%5E2+%2B16ci+%2B16%29+ ALIGN=MIDDLE  ALT=\"+%28c-2i%29%28c%5E4+-8c%5E3i+%2B8c%5E2+%2B16ci+%2B16%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28c%2Ac%5E4%29+%2B+%28c%2A-8c%5E3i%29+%2B+%28c%2A8c%5E2%29+%2B+%28c%2A16ci%29+%2B+%28c%2A16%29+%2B+%28-2i%2Ac%5E4%29%2B%28-2i%2A-8c%5E3i%29%2B+%28-2i%2A8c%5E2%29+%2B+%28-2i%2A16ci%29+%2B+%28-2i%2A16%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply, working the i^2 as well<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c%5E5+-8c%5E4i+%2B8c%5E3+%2B16c%5E2i+%2B16c+-2c%5E4i+%2B16c%5E3%28-1%29+-16c%5E2i+-32c%28-1%29+-32i+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "combine like terms<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c%5E5+-10c%5E4i+-+8c%5E3i+%2B+48c+-+32i+ ALIGN=MIDDLE  ALT=\"+c%5E5+-10c%5E4i+-+8c%5E3i+%2B+48c+-+32i+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the c^2 cancel out\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );