document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=403073>Question 403073</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, can you please help me?\r<BR>\n" );
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document.write( "I need to factor the following completely:\r<BR>\n" );
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document.write( "21x^3-18x^2y+24xy^2<BR>\n" );
document.write( "Can you please show me how?<BR>\n" );
document.write( "thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #285115 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=285115\">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=21x%5E3-18x%5E2y%2B24xy%5E2 ALIGN=MIDDLE  ALT=\"21x%5E3-18x%5E2y%2B24xy%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When factoring, always start with the Greatest Common Factor (GCF). The GCF here is 3x:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%287x%5E2-6xy%2B8y%5E2%29 ALIGN=MIDDLE  ALT=\"3x%287x%5E2-6xy%2B8y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Always keep factoring until no more factoring can be done. The second factor is a trinomial we should try to factor. It does not fit any of the patterns for factoring. But we can try trinomial factoring. This factoring is \"un-FOIL-ing\" the expression. We are looking to see if some pair of binomials will result in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%5E2-6xy%2B8y%5E2 ALIGN=MIDDLE  ALT=\"7x%5E2-6xy%2B8y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > when multiplied. We are looking for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%5E2-6xy%2B8y%5E2+=+%28a%5B1%5D+%2B+b%5B1%5D%29%28a%5B2%5D+%2B+b%5B2%5D%29 ALIGN=MIDDLE  ALT=\"7x%5E2-6xy%2B8y%5E2+=+%28a%5B1%5D+%2B+b%5B1%5D%29%28a%5B2%5D+%2B+b%5B2%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D ALIGN=MIDDLE  ALT=\"a%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the \"First\" terms) are factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%5E2 ALIGN=MIDDLE  ALT=\"7x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D ALIGN=MIDDLE  ALT=\"b%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B2%5D ALIGN=MIDDLE  ALT=\"b%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the \"Last\" terms) are factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E2 ALIGN=MIDDLE  ALT=\"8y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. That is the easy part the hard part is that the \"Outside\" terms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B2%5D ALIGN=MIDDLE  ALT=\"b%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and \"Inside\" terms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D ALIGN=MIDDLE  ALT=\"a%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D ALIGN=MIDDLE  ALT=\"b%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > must produce the -6xy if the expression is to factor at all.<br><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D ALIGN=MIDDLE  ALT=\"a%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > there is really only one choice: 7x and x. For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D ALIGN=MIDDLE  ALT=\"b%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B2%5D ALIGN=MIDDLE  ALT=\"b%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > there are several choices. Since the middle term has a minus in front of it we need them both to be negative. But they could be -8y and -y or -2y and -4y. And which one is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D ALIGN=MIDDLE  ALT=\"b%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and which one is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B2%5D ALIGN=MIDDLE  ALT=\"b%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also not known yet.<br><BR>\n" );
document.write( "So the possibilities are:<BR>\n" );
document.write( "(7x-8y)(x-y)<BR>\n" );
document.write( "(7x-y)(x-8y)<BR>\n" );
document.write( "(7x-2y)(x-4y)<BR>\n" );
document.write( "or<BR>\n" );
document.write( "(7x-4y)(x-2y)<BR>\n" );
document.write( "All of these will produce the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%5E2 ALIGN=MIDDLE  ALT=\"7x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8y%5E2 ALIGN=MIDDLE  ALT=\"8y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. But only 1 (or none) of these will produce the -6xy. And that will come from the Outside and Inside terms. Let's try each one, looking at just the Outside and Inside multiplications of FOIL:<BR>\n" );
document.write( "(7x-8y)(x-y)<BR>\n" );
document.write( "Outside: 7x*(-y) = -7xy<BR>\n" );
document.write( "Inside: x*(-8y)  = -8xy<BR>\n" );
document.write( "Together they make -15xy<br><BR>\n" );
document.write( "(7x-y)(x-8y)<BR>\n" );
document.write( "Outside: 7x*(-8y) = -56xy<BR>\n" );
document.write( "Inside: x*(-y)  = -xy<BR>\n" );
document.write( "Together they make -57xy<br><BR>\n" );
document.write( "(7x-2y)(x-4y)<BR>\n" );
document.write( "Outside: 7x*(-4y) = -28xy<BR>\n" );
document.write( "Inside: x*(-2y)  = -2xy<BR>\n" );
document.write( "Together they make -30xy<br><BR>\n" );
document.write( "(7x-4y)(x-2y)<BR>\n" );
document.write( "Outside: 7x*(-2y) = -14xy<BR>\n" );
document.write( "Inside: x*(-4y)  = -4xy<BR>\n" );
document.write( "Together they make -18xy<br><BR>\n" );
document.write( "None of these produced the -6xy we were looking for. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7x%5E2-6xy%2B8y%5E2 ALIGN=MIDDLE  ALT=\"7x%5E2-6xy%2B8y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > will not factor any further. So your fully factored expression is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%287x%5E2-6xy%2B8y%5E2%29 ALIGN=MIDDLE  ALT=\"3x%287x%5E2-6xy%2B8y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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