document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=688801>Question 688801</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having trouble understanding how my instructor got this answer from<BR>\n" );
document.write( "factoring this problem.\r<BR>\n" );
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document.write( "125c^6 - 8d^6<BR>\n" );
document.write( "=a^3-b^3= (a-b)*(a^2+ab+b^2)\r<BR>\n" );
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document.write( "Can you please explain what he did? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #425701 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=425701\">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=a%5E3-b%5E3=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"a%5E3-b%5E3=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not the answer.<BR>\n" );
document.write( "It's more like the procedure to get the answer.<BR>\n" );
document.write( "It's a formula affectionately called \"difference of two cubes\".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3-b%5E3=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"a%5E3-b%5E3=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29=a%5E3-b%5E3 ALIGN=MIDDLE  ALT=\"%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29=a%5E3-b%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "is one of those \"special products\"<BR>\n" );
document.write( "teachers and textbooks list.<BR>\n" );
document.write( "Those special products \"formulas\" are true if you substitute<BR>\n" );
document.write( "any number or expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and any number or expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "In the case of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125c%5E6+-+8d%5E6 ALIGN=MIDDLE  ALT=\"125c%5E6+-+8d%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "you are expected to notice that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125c%5E6=5%5E3%2A%28c%5E2%29%5E3=%285c%5E2%29%5E3 ALIGN=MIDDLE  ALT=\"125c%5E6=5%5E3%2A%28c%5E2%29%5E3=%285c%5E2%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so your <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5c%5E2 ALIGN=MIDDLE  ALT=\"5c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You are also expected to notice that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5E6=2%5E3%2A%28d%5E2%29%5E3=%282d%5E2%29%5E3 ALIGN=MIDDLE  ALT=\"8d%5E6=2%5E3%2A%28d%5E2%29%5E3=%282d%5E2%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so your <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2d%5E2 ALIGN=MIDDLE  ALT=\"2d%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting your expressions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3-b%5E3=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"a%5E3-b%5E3=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > you get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125c%5E6-8d%5E6=%285c%5E2-2d%5E2%29%2A%28%285c%5E2%29%5E2%2B%285c%5E2%29%282d%5E2%29%2B%282d%5E2%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can go a little further, working on the right hand side and write it as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285c%5E2-2d%5E2%29%2A%2825c%5E4%2B10c%5E2d%5E2%2B4d%5E4%29 ALIGN=MIDDLE  ALT=\"%285c%5E2-2d%5E2%29%2A%2825c%5E4%2B10c%5E2d%5E2%2B4d%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A little more can be done, but it gets ugly.<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29=a%5E3-b%5E3 ALIGN=MIDDLE  ALT=\"%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29=a%5E3-b%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not a magical formula.<BR>\n" );
document.write( "You can easily prove it if you multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%2A%28a%5E2%2Bab%2Bb%5E2%29=a%28a%5E2%2Bab%2Bb%5E2%29-b%28a%5E2%2Bab%2Bb%5E2%29=a%5E3%2Ba%5E2b%2Bab%5E2-%28a%5Eb%2Bab%5E2%2Bb%5E3%29=a%5E3%2Ba%5E2b%2Bab%5E2-a%5Eb-ab%5E2-b%5E3=a%5E3-b%5E3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "EXTRA TIP:<BR>\n" );
document.write( "Other special products are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3%2Bb%5E3=%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"a%5E3%2Bb%5E3=%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29=a%5E3%2Bb%5E3 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28a%5E2-ab%2Bb%5E2%29=a%5E3%2Bb%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (sum of two cubes)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-b%5E2=%28a%2Bb%29%28a-b%29 ALIGN=MIDDLE  ALT=\"a%5E2-b%5E2=%28a%2Bb%29%28a-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28a-b%29=a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28a-b%29=a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > difference of two squares<BR>\n" );
document.write( "and also<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (square of a binomial)<BR>\n" );
document.write( " <BR>\n" );
document.write( "RANT:<BR>\n" );
document.write( "This part of math (mulltiplying and factoring polynomials) is a nightmare for most students.<BR>\n" );
document.write( "Unfortunately, factoring polynomials will keep coming back at various points through your current course and all the math courses that follow.<BR>\n" );
document.write( "(Rational functions is one example).<BR>\n" );
document.write( "It is useful to remember the special products formulas.<BR>\n" );
document.write( "It is easier to remember them if you understand what they mean.<BR>\n" );
document.write( "(The cute, affectionate names help).<BR>\n" );
document.write( "And lots of practice with polynomials helps.<BR>\n" );
document.write( "I was never a fan of homework, and rarely needed to do much practice with math, but I found out the hard way that this part of math required work. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );