document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=115668>Question 115668</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor completely. m cubed - 9m squared n+ 18n squared m </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84173 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=84173\">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=m%5E3-9m%5E2n%2B18n%5E2m ALIGN=MIDDLE  ALT=\"m%5E3-9m%5E2n%2B18n%5E2m\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28m%5E2-9mn%2B18n%5E2%29 ALIGN=MIDDLE  ALT=\"m%28m%5E2-9mn%2B18n%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's focus on the inner expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-9mn%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-9mn%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-9mn%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-9mn%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2 ALIGN=MIDDLE  ALT=\"m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18n%5E2 ALIGN=MIDDLE  ALT=\"18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and 18 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient 18 to get 18. Now what two numbers multiply to 18 and add to the  middle coefficient -9? Let's list all of the factors of 18:\r<BR>\n" );
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document.write( "Factors of 18:\r<BR>\n" );
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document.write( "1,2,3,6,9,18\r<BR>\n" );
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document.write( "-1,-2,-3,-6,-9,-18 ...List the negative factors as well. This will allow us to find all possible combinations\r<BR>\n" );
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document.write( "These factors pair up and multiply to 18\r<BR>\n" );
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document.write( "1*18\r<BR>\n" );
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document.write( "2*9\r<BR>\n" );
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document.write( "3*6\r<BR>\n" );
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document.write( "(-1)*(-18)\r<BR>\n" );
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document.write( "(-2)*(-9)\r<BR>\n" );
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document.write( "(-3)*(-6)\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">18</TD><TD >1+18=19</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">9</TD><TD >2+9=11</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">6</TD><TD >3+6=9</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-18</TD><TD >-1+(-18)=-19</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-9</TD><TD >-2+(-9)=-11</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-6</TD><TD >-3+(-6)=-9</TD></tr>\n" );
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document.write( "From this list we can see that -3 and -6 add up to -9 and multiply to 18\r<BR>\n" );
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document.write( "Now looking at the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-9mn%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-9mn%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9mn ALIGN=MIDDLE  ALT=\"-9mn\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3mn%2B-6mn ALIGN=MIDDLE  ALT=\"-3mn%2B-6mn\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3mn%2B-6mn ALIGN=MIDDLE  ALT=\"-3mn%2B-6mn\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9mn ALIGN=MIDDLE  ALT=\"-9mn\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9mn ALIGN=MIDDLE  ALT=\"-9mn\"   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=m%5E2%2Bhighlight%28-3mn%2B-6mn%29%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2%2Bhighlight%28-3mn%2B-6mn%29%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-3mn-6mn%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-3mn-6mn%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > by grouping:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%5E2-3mn%29%2B%28-6mn%2B18n%5E2%29 ALIGN=MIDDLE  ALT=\"%28m%5E2-3mn%29%2B%28-6mn%2B18n%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Group like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28m-3n%29-6n%28m-3n%29 ALIGN=MIDDLE  ALT=\"m%28m-3n%29-6n%28m-3n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the first group. Factor out the GCF of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6n ALIGN=MIDDLE  ALT=\"-6n\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of the second group\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m-6n%29%28m-3n%29 ALIGN=MIDDLE  ALT=\"%28m-6n%29%28m-3n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since we have a common term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m-3n ALIGN=MIDDLE  ALT=\"m-3n\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can combine like terms\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-3mn-6mn%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-3mn-6mn%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m-6n%29%28m-3n%29 ALIGN=MIDDLE  ALT=\"%28m-6n%29%28m-3n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this also means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-9mn%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-9mn%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m-6n%29%28m-3n%29 ALIGN=MIDDLE  ALT=\"%28m-6n%29%28m-3n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-9mn%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-9mn%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-3mn-6mn%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-3mn-6mn%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-9mn%2B18n%5E2 ALIGN=MIDDLE  ALT=\"m%5E2-9mn%2B18n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m-6n%29%28m-3n%29 ALIGN=MIDDLE  ALT=\"%28m-6n%29%28m-3n%29\"   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=m%28m-6n%29%28m-3n%29 ALIGN=MIDDLE  ALT=\"m%28m-6n%29%28m-3n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now reintroduce the GCF\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E3-9m%5E2n%2B18n%5E2m ALIGN=MIDDLE  ALT=\"m%5E3-9m%5E2n%2B18n%5E2m\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%28m-6n%29%28m-3n%29 ALIGN=MIDDLE  ALT=\"m%28m-6n%29%28m-3n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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