document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=187134>Question 187134</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please factor out this polynomial.<BR>\n" );
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document.write( "4x+12xy+9y^2<BR>\n" );
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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 #140274 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=140274\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm assuming that the expression is really <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B12xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B12xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If not, then let me know.\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B12xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B12xy%2B9y%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=4x%5E2 ALIGN=MIDDLE  ALT=\"4x%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=9y%5E2 ALIGN=MIDDLE  ALT=\"9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 4 and 9 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 4 and the last coefficient 9 to get 36. Now what two numbers multiply to 36 and add to the  middle coefficient 12? Let's list all of the factors of 36:\r<BR>\n" );
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document.write( "Factors of 36:\r<BR>\n" );
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document.write( "1,2,3,4,6,9,12,18\r<BR>\n" );
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document.write( "-1,-2,-3,-4,-6,-9,-12,-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 36\r<BR>\n" );
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document.write( "1*36\r<BR>\n" );
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document.write( "2*18\r<BR>\n" );
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document.write( "3*12\r<BR>\n" );
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document.write( "4*9\r<BR>\n" );
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document.write( "6*6\r<BR>\n" );
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document.write( "(-1)*(-36)\r<BR>\n" );
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document.write( "(-2)*(-18)\r<BR>\n" );
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document.write( "(-3)*(-12)\r<BR>\n" );
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document.write( "(-4)*(-9)\r<BR>\n" );
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document.write( "(-6)*(-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 12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 12\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\">36</TD><TD >1+36=37</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">18</TD><TD >2+18=20</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">12</TD><TD >3+12=15</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">9</TD><TD >4+9=13</TD></tr><tr><TD  align=\"center\">6</TD><TD  align=\"center\">6</TD><TD >6+6=12</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-36</TD><TD >-1+(-36)=-37</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-18</TD><TD >-2+(-18)=-20</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-12</TD><TD >-3+(-12)=-15</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">-9</TD><TD >-4+(-9)=-13</TD></tr><tr><TD  align=\"center\">-6</TD><TD  align=\"center\">-6</TD><TD >-6+(-6)=-12</TD></tr>\n" );
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document.write( "From this list we can see that 6 and 6 add up to 12 and multiply to 36\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=4x%5E2%2B12xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B12xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12xy ALIGN=MIDDLE  ALT=\"12xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6xy%2B6xy ALIGN=MIDDLE  ALT=\"6xy%2B6xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6xy%2B6xy ALIGN=MIDDLE  ALT=\"6xy%2B6xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12xy ALIGN=MIDDLE  ALT=\"12xy\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12xy ALIGN=MIDDLE  ALT=\"12xy\"   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=4x%5E2%2Bhighlight%286xy%2B6xy%29%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2Bhighlight%286xy%2B6xy%29%2B9y%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=4x%5E2%2B6xy%2B6xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B6xy%2B6xy%2B9y%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=%284x%5E2%2B6xy%29%2B%286xy%2B9y%5E2%29 ALIGN=MIDDLE  ALT=\"%284x%5E2%2B6xy%29%2B%286xy%2B9y%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=2x%282x%2B3y%29%2B3y%282x%2B3y%29 ALIGN=MIDDLE  ALT=\"2x%282x%2B3y%29%2B3y%282x%2B3y%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=2x ALIGN=MIDDLE  ALT=\"2x\"   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=3y ALIGN=MIDDLE  ALT=\"3y\"   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=%282x%2B3y%29%282x%2B3y%29 ALIGN=MIDDLE  ALT=\"%282x%2B3y%29%282x%2B3y%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=2x%2B3y ALIGN=MIDDLE  ALT=\"2x%2B3y\"   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=4x%5E2%2B6xy%2B6xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B6xy%2B6xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B3y%29%282x%2B3y%29 ALIGN=MIDDLE  ALT=\"%282x%2B3y%29%282x%2B3y%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=4x%5E2%2B12xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B12xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B3y%29%282x%2B3y%29 ALIGN=MIDDLE  ALT=\"%282x%2B3y%29%282x%2B3y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B12xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B12xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B6xy%2B6xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B6xy%2B6xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "note:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B3y%29%282x%2B3y%29 ALIGN=MIDDLE  ALT=\"%282x%2B3y%29%282x%2B3y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B3y%29%5E2 ALIGN=MIDDLE  ALT=\"%282x%2B3y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B3y ALIGN=MIDDLE  ALT=\"2x%2B3y\"   DISABLED_style= \"max-width:90%; height:auto;\" > occurs twice. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B12xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B12xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > also factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B3y%29%5E2 ALIGN=MIDDLE  ALT=\"%282x%2B3y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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=4x%5E2%2B12xy%2B9y%5E2 ALIGN=MIDDLE  ALT=\"4x%5E2%2B12xy%2B9y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B3y%29%5E2 ALIGN=MIDDLE  ALT=\"%282x%2B3y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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