document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=115002>Question 115002</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor completely.\r<BR>\n" );
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document.write( "2p squared + 11p + 12 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83673 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=83673\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p%5E2%2B11p%2B12 ALIGN=MIDDLE  ALT=\"2p%5E2%2B11p%2B12\"   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=2p%5E2 ALIGN=MIDDLE  ALT=\"2p%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=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 2 and 12 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 2 and the last coefficient 12 to get 24. Now what two numbers multiply to 24 and add to the  middle coefficient 11? Let's list all of the factors of 24:\r<BR>\n" );
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document.write( "Factors of 24:\r<BR>\n" );
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document.write( "1,2,3,4,6,8,12,24\r<BR>\n" );
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document.write( "-1,-2,-3,-4,-6,-8,-12,-24 ...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 24\r<BR>\n" );
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document.write( "1*24\r<BR>\n" );
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document.write( "2*12\r<BR>\n" );
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document.write( "3*8\r<BR>\n" );
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document.write( "4*6\r<BR>\n" );
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document.write( "(-1)*(-24)\r<BR>\n" );
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document.write( "(-2)*(-12)\r<BR>\n" );
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document.write( "(-3)*(-8)\r<BR>\n" );
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document.write( "(-4)*(-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 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11\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\">24</TD><TD >1+24=25</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">12</TD><TD >2+12=14</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">8</TD><TD >3+8=11</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">6</TD><TD >4+6=10</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-24</TD><TD >-1+(-24)=-25</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-12</TD><TD >-2+(-12)=-14</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-8</TD><TD >-3+(-8)=-11</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">-6</TD><TD >-4+(-6)=-10</TD></tr>\n" );
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document.write( "From this list we can see that 3 and 8 add up to 11 and multiply to 24\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=2p%5E2%2B11p%2B12 ALIGN=MIDDLE  ALT=\"2p%5E2%2B11p%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11p ALIGN=MIDDLE  ALT=\"11p\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p%2B8p ALIGN=MIDDLE  ALT=\"3p%2B8p\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p%2B8p ALIGN=MIDDLE  ALT=\"3p%2B8p\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11p ALIGN=MIDDLE  ALT=\"11p\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11p ALIGN=MIDDLE  ALT=\"11p\"   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=2p%5E2%2Bhighlight%283p%2B8p%29%2B12 ALIGN=MIDDLE  ALT=\"2p%5E2%2Bhighlight%283p%2B8p%29%2B12\"   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=2p%5E2%2B3p%2B8p%2B12 ALIGN=MIDDLE  ALT=\"2p%5E2%2B3p%2B8p%2B12\"   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=%282p%5E2%2B3p%29%2B%288p%2B12%29 ALIGN=MIDDLE  ALT=\"%282p%5E2%2B3p%29%2B%288p%2B12%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=p%282p%2B3%29%2B4%282p%2B3%29 ALIGN=MIDDLE  ALT=\"p%282p%2B3%29%2B4%282p%2B3%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=p ALIGN=MIDDLE  ALT=\"p\"   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=4 ALIGN=MIDDLE  ALT=\"4\"   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=%28p%2B4%29%282p%2B3%29 ALIGN=MIDDLE  ALT=\"%28p%2B4%29%282p%2B3%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=2p%2B3 ALIGN=MIDDLE  ALT=\"2p%2B3\"   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=2p%5E2%2B3p%2B8p%2B12 ALIGN=MIDDLE  ALT=\"2p%5E2%2B3p%2B8p%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28p%2B4%29%282p%2B3%29 ALIGN=MIDDLE  ALT=\"%28p%2B4%29%282p%2B3%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=2p%5E2%2B11p%2B12 ALIGN=MIDDLE  ALT=\"2p%5E2%2B11p%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28p%2B4%29%282p%2B3%29 ALIGN=MIDDLE  ALT=\"%28p%2B4%29%282p%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p%5E2%2B11p%2B12 ALIGN=MIDDLE  ALT=\"2p%5E2%2B11p%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p%5E2%2B3p%2B8p%2B12 ALIGN=MIDDLE  ALT=\"2p%5E2%2B3p%2B8p%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( " </SPAN>\n" );
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