document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=114213>Question 114213</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor the trinomial completely.\r<BR>\n" );
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document.write( "2a to the 3rd power - 52a squared b + 96ab squared </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83127 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=83127\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E3-52a%5E2b%2B96ab%5E2 ALIGN=MIDDLE  ALT=\"2a%5E3-52a%5E2b%2B96ab%5E2\"   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=2a%28a%5E2-26ab%2B48b%5E2%29 ALIGN=MIDDLE  ALT=\"2a%28a%5E2-26ab%2B48b%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=2a ALIGN=MIDDLE  ALT=\"2a\"   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=a%5E2-26ab%2B48b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-26ab%2B48b%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=a%5E2-26ab%2B48b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-26ab%2B48b%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=a%5E2 ALIGN=MIDDLE  ALT=\"a%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=48b%5E2 ALIGN=MIDDLE  ALT=\"48b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and 48 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient 48 to get 48. Now what two numbers multiply to 48 and add to the  middle coefficient -26? Let's list all of the factors of 48:\r<BR>\n" );
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document.write( "Factors of 48:\r<BR>\n" );
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document.write( "1,2,3,4,6,8,12,16,24,48\r<BR>\n" );
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document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...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 48\r<BR>\n" );
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document.write( "1*48\r<BR>\n" );
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document.write( "2*24\r<BR>\n" );
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document.write( "3*16\r<BR>\n" );
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document.write( "4*12\r<BR>\n" );
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document.write( "6*8\r<BR>\n" );
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document.write( "(-1)*(-48)\r<BR>\n" );
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document.write( "(-2)*(-24)\r<BR>\n" );
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document.write( "(-3)*(-16)\r<BR>\n" );
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document.write( "(-4)*(-12)\r<BR>\n" );
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document.write( "(-6)*(-8)\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 -26? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -26\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\">48</TD><TD >1+48=49</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">24</TD><TD >2+24=26</TD></tr><tr><TD  align=\"center\">3</TD><TD  align=\"center\">16</TD><TD >3+16=19</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">12</TD><TD >4+12=16</TD></tr><tr><TD  align=\"center\">6</TD><TD  align=\"center\">8</TD><TD >6+8=14</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-48</TD><TD >-1+(-48)=-49</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-24</TD><TD >-2+(-24)=-26</TD></tr><tr><TD  align=\"center\">-3</TD><TD  align=\"center\">-16</TD><TD >-3+(-16)=-19</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">-12</TD><TD >-4+(-12)=-16</TD></tr><tr><TD  align=\"center\">-6</TD><TD  align=\"center\">-8</TD><TD >-6+(-8)=-14</TD></tr>\n" );
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document.write( "From this list we can see that -2 and -24 add up to -26 and multiply to 48\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=a%5E2-26ab%2B48b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-26ab%2B48b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-26ab ALIGN=MIDDLE  ALT=\"-26ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2ab%2B-24ab ALIGN=MIDDLE  ALT=\"-2ab%2B-24ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2ab%2B-24ab ALIGN=MIDDLE  ALT=\"-2ab%2B-24ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-26ab ALIGN=MIDDLE  ALT=\"-26ab\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-26ab ALIGN=MIDDLE  ALT=\"-26ab\"   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=a%5E2%2Bhighlight%28-2ab%2B-24ab%29%2B48b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2%2Bhighlight%28-2ab%2B-24ab%29%2B48b%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=a%5E2-2ab-24ab%2B48b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-2ab-24ab%2B48b%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=%28a%5E2-2ab%29%2B%28-24ab%2B48b%5E2%29 ALIGN=MIDDLE  ALT=\"%28a%5E2-2ab%29%2B%28-24ab%2B48b%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=a%28a-2b%29-24b%28a-2b%29 ALIGN=MIDDLE  ALT=\"a%28a-2b%29-24b%28a-2b%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=a ALIGN=MIDDLE  ALT=\"a\"   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=-24b ALIGN=MIDDLE  ALT=\"-24b\"   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=%28a-24b%29%28a-2b%29 ALIGN=MIDDLE  ALT=\"%28a-24b%29%28a-2b%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=a-2b ALIGN=MIDDLE  ALT=\"a-2b\"   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=a%5E2-2ab-24ab%2B48b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-2ab-24ab%2B48b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-24b%29%28a-2b%29 ALIGN=MIDDLE  ALT=\"%28a-24b%29%28a-2b%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=a%5E2-26ab%2B48b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-26ab%2B48b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-24b%29%28a-2b%29 ALIGN=MIDDLE  ALT=\"%28a-24b%29%28a-2b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-26ab%2B48b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-26ab%2B48b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-2ab-24ab%2B48b%5E2 ALIGN=MIDDLE  ALT=\"a%5E2-2ab-24ab%2B48b%5E2\"   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=2a%28a-24b%29%28a-2b%29 ALIGN=MIDDLE  ALT=\"2a%28a-24b%29%28a-2b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >Now reintroduce the GCF back in\r<BR>\n" );
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document.write( "==============================\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=2a%5E3-52a%5E2b%2B96ab%5E2 ALIGN=MIDDLE  ALT=\"2a%5E3-52a%5E2b%2B96ab%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%28a-24b%29%28a-2b%29 ALIGN=MIDDLE  ALT=\"2a%28a-24b%29%28a-2b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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