document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=132112>Question 132112</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2%2B18s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B18s%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96468 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=96468\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2%2B18s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B18s%2B81\"   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=s%5E2 ALIGN=MIDDLE  ALT=\"s%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=81 ALIGN=MIDDLE  ALT=\"81\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and 81 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient 81 to get 81. Now what two numbers multiply to 81 and add to the  middle coefficient 18? Let's list all of the factors of 81:\r<BR>\n" );
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document.write( "Factors of 81:\r<BR>\n" );
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document.write( "1,3,9,27\r<BR>\n" );
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document.write( "-1,-3,-9,-27 ...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 81\r<BR>\n" );
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document.write( "1*81\r<BR>\n" );
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document.write( "3*27\r<BR>\n" );
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document.write( "9*9\r<BR>\n" );
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document.write( "(-1)*(-81)\r<BR>\n" );
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document.write( "(-3)*(-27)\r<BR>\n" );
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document.write( "(-9)*(-9)\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 18? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 18\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\">81</TD><TD >1+81=82</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">27</TD><TD >3+27=30</TD></tr><tr><TD  align=\"center\">9</TD><TD  align=\"center\">9</TD><TD >9+9=18</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-81</TD><TD >-1+(-81)=-82</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-27</TD><TD >-3+(-27)=-30</TD></tr><tr><TD  align=\"center\">-9</TD><TD  align=\"center\">-9</TD><TD >-9+(-9)=-18</TD></tr>\n" );
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document.write( "From this list we can see that 9 and 9 add up to 18 and multiply to 81\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=s%5E2%2B18s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B18s%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18s ALIGN=MIDDLE  ALT=\"18s\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9s%2B9s ALIGN=MIDDLE  ALT=\"9s%2B9s\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9s%2B9s ALIGN=MIDDLE  ALT=\"9s%2B9s\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18s ALIGN=MIDDLE  ALT=\"18s\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18s ALIGN=MIDDLE  ALT=\"18s\"   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=s%5E2%2Bhighlight%289s%2B9s%29%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2Bhighlight%289s%2B9s%29%2B81\"   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=s%5E2%2B9s%2B9s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B9s%2B9s%2B81\"   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=%28s%5E2%2B9s%29%2B%289s%2B81%29 ALIGN=MIDDLE  ALT=\"%28s%5E2%2B9s%29%2B%289s%2B81%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=s%28s%2B9%29%2B9%28s%2B9%29 ALIGN=MIDDLE  ALT=\"s%28s%2B9%29%2B9%28s%2B9%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=s ALIGN=MIDDLE  ALT=\"s\"   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=9 ALIGN=MIDDLE  ALT=\"9\"   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=%28s%2B9%29%28s%2B9%29 ALIGN=MIDDLE  ALT=\"%28s%2B9%29%28s%2B9%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=s%2B9 ALIGN=MIDDLE  ALT=\"s%2B9\"   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=s%5E2%2B9s%2B9s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B9s%2B9s%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28s%2B9%29%28s%2B9%29 ALIGN=MIDDLE  ALT=\"%28s%2B9%29%28s%2B9%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=s%5E2%2B18s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B18s%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28s%2B9%29%28s%2B9%29 ALIGN=MIDDLE  ALT=\"%28s%2B9%29%28s%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2%2B18s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B18s%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2%2B9s%2B9s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B9s%2B9s%2B81\"   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=%28s%2B9%29%28s%2B9%29 ALIGN=MIDDLE  ALT=\"%28s%2B9%29%28s%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28s%2B9%29%5E2 ALIGN=MIDDLE  ALT=\"%28s%2B9%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%2B9 ALIGN=MIDDLE  ALT=\"s%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" > occurs twice. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2%2B18s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B18s%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" > also factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28s%2B9%29%5E2 ALIGN=MIDDLE  ALT=\"%28s%2B9%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=s%5E2%2B18s%2B81 ALIGN=MIDDLE  ALT=\"s%5E2%2B18s%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28s%2B9%29%5E2 ALIGN=MIDDLE  ALT=\"%28s%2B9%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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