document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=191469>Question 191469</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help! Can't find any examples of these in my book, there's always a different exponent or something....\r<BR>\n" );
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document.write( "Factor the polynomial\r<BR>\n" );
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document.write( "1.) (32k^3 n^2 m)+(112k^2 nm^2)+(98km^3)\r<BR>\n" );
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document.write( "2.) (9k^2)-(16m^2)\r<BR>\n" );
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document.write( "3.) (147a^4 b)-(48b^3)\r<BR>\n" );
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document.write( "4.) (x^2)-(2/3)x+c\r<BR>\n" );
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document.write( "I know I asked more than one but I really need these answers asap I will not ask anymore today. Also I'm on an iPhone and it's already a pain to text on it I couldn't imgine going back and forth on a website. Thank you so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #143683 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=143683\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first two to get you started, repost if you need more help.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32k%5E3n%5E2m%2B112k%5E2nm%5E2%2B98km%5E3 ALIGN=MIDDLE  ALT=\"32k%5E3n%5E2m%2B112k%5E2nm%5E2%2B98km%5E3\"   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=2km%2816k%5E2n%5E2%2B56knm%2B49m%5E2%29 ALIGN=MIDDLE  ALT=\"2km%2816k%5E2n%5E2%2B56knm%2B49m%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=2km ALIGN=MIDDLE  ALT=\"2km\"   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=16k%5E2n%5E2%2B56knm%2B49m%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%5E2%2B56knm%2B49m%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=16k%5E2n%5E2%2B56knm%2B49m%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%5E2%2B56knm%2B49m%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=16k%5E2n%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%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=49m%5E2 ALIGN=MIDDLE  ALT=\"49m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 16 and 49 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 16 and the last coefficient 49 to get 784. Now what two numbers multiply to 784 and add to the  middle coefficient 56? Let's list all of the factors of 784:\r<BR>\n" );
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document.write( "Factors of 784:\r<BR>\n" );
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document.write( "1,2,4,7,8,14,16,28,49,56,98,112,196,392\r<BR>\n" );
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document.write( "-1,-2,-4,-7,-8,-14,-16,-28,-49,-56,-98,-112,-196,-392 ...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 784\r<BR>\n" );
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document.write( "1*784\r<BR>\n" );
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document.write( "2*392\r<BR>\n" );
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document.write( "4*196\r<BR>\n" );
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document.write( "7*112\r<BR>\n" );
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document.write( "8*98\r<BR>\n" );
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document.write( "14*56\r<BR>\n" );
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document.write( "16*49\r<BR>\n" );
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document.write( "28*28\r<BR>\n" );
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document.write( "(-1)*(-784)\r<BR>\n" );
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document.write( "(-2)*(-392)\r<BR>\n" );
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document.write( "(-4)*(-196)\r<BR>\n" );
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document.write( "(-7)*(-112)\r<BR>\n" );
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document.write( "(-8)*(-98)\r<BR>\n" );
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document.write( "(-14)*(-56)\r<BR>\n" );
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document.write( "(-16)*(-49)\r<BR>\n" );
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document.write( "(-28)*(-28)\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 56? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 56\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\">784</TD><TD >1+784=785</TD></tr><tr><TD  align=\"center\">2</TD><TD  align=\"center\">392</TD><TD >2+392=394</TD></tr><tr><TD  align=\"center\">4</TD><TD  align=\"center\">196</TD><TD >4+196=200</TD></tr><tr><TD  align=\"center\">7</TD><TD  align=\"center\">112</TD><TD >7+112=119</TD></tr><tr><TD  align=\"center\">8</TD><TD  align=\"center\">98</TD><TD >8+98=106</TD></tr><tr><TD  align=\"center\">14</TD><TD  align=\"center\">56</TD><TD >14+56=70</TD></tr><tr><TD  align=\"center\">16</TD><TD  align=\"center\">49</TD><TD >16+49=65</TD></tr><tr><TD  align=\"center\">28</TD><TD  align=\"center\">28</TD><TD >28+28=56</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-784</TD><TD >-1+(-784)=-785</TD></tr><tr><TD  align=\"center\">-2</TD><TD  align=\"center\">-392</TD><TD >-2+(-392)=-394</TD></tr><tr><TD  align=\"center\">-4</TD><TD  align=\"center\">-196</TD><TD >-4+(-196)=-200</TD></tr><tr><TD  align=\"center\">-7</TD><TD  align=\"center\">-112</TD><TD >-7+(-112)=-119</TD></tr><tr><TD  align=\"center\">-8</TD><TD  align=\"center\">-98</TD><TD >-8+(-98)=-106</TD></tr><tr><TD  align=\"center\">-14</TD><TD  align=\"center\">-56</TD><TD >-14+(-56)=-70</TD></tr><tr><TD  align=\"center\">-16</TD><TD  align=\"center\">-49</TD><TD >-16+(-49)=-65</TD></tr><tr><TD  align=\"center\">-28</TD><TD  align=\"center\">-28</TD><TD >-28+(-28)=-56</TD></tr>\n" );
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document.write( "From this list we can see that 28 and 28 add up to 56 and multiply to 784\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=16k%5E2n%5E2%2B56knm%2B49m%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%5E2%2B56knm%2B49m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=56knm ALIGN=MIDDLE  ALT=\"56knm\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28knm%2B28knm ALIGN=MIDDLE  ALT=\"28knm%2B28knm\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28knm%2B28knm ALIGN=MIDDLE  ALT=\"28knm%2B28knm\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=56knm ALIGN=MIDDLE  ALT=\"56knm\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=56knm ALIGN=MIDDLE  ALT=\"56knm\"   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=16k%5E2n%5E2%2Bhighlight%2828knm%2B28knm%29%2B49m%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%5E2%2Bhighlight%2828knm%2B28knm%29%2B49m%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=16k%5E2n%5E2%2B28knm%2B28knm%2B49m%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%5E2%2B28knm%2B28knm%2B49m%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=%2816k%5E2n%5E2%2B28knm%29%2B%2828knm%2B49m%5E2%29 ALIGN=MIDDLE  ALT=\"%2816k%5E2n%5E2%2B28knm%29%2B%2828knm%2B49m%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=4kn%284kn%2B7m%29%2B7m%284kn%2B7m%29 ALIGN=MIDDLE  ALT=\"4kn%284kn%2B7m%29%2B7m%284kn%2B7m%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=4kn ALIGN=MIDDLE  ALT=\"4kn\"   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=7m ALIGN=MIDDLE  ALT=\"7m\"   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=%284kn%2B7m%29%284kn%2B7m%29 ALIGN=MIDDLE  ALT=\"%284kn%2B7m%29%284kn%2B7m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284kn%2B7m%29%5E2 ALIGN=MIDDLE  ALT=\"%284kn%2B7m%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Condense the terms.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16k%5E2n%5E2%2B28knm%2B28knm%2B49m%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%5E2%2B28knm%2B28knm%2B49m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284kn%2B7m%29%5E2 ALIGN=MIDDLE  ALT=\"%284kn%2B7m%29%5E2\"   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=16k%5E2n%5E2%2B56knm%2B49m%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%5E2%2B56knm%2B49m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284kn%2B7m%29%5E2 ALIGN=MIDDLE  ALT=\"%284kn%2B7m%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16k%5E2n%5E2%2B56knm%2B49m%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%5E2%2B56knm%2B49m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16k%5E2n%5E2%2B28knm%2B28knm%2B49m%5E2 ALIGN=MIDDLE  ALT=\"16k%5E2n%5E2%2B28knm%2B28knm%2B49m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "So our expression goes from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2km%2816k%5E2n%5E2%2B56knm%2B49m%5E2%29 ALIGN=MIDDLE  ALT=\"2km%2816k%5E2n%5E2%2B56knm%2B49m%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and factors further to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2km%284kn%2B7m%29%5E2 ALIGN=MIDDLE  ALT=\"2km%284kn%2B7m%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=32k%5E3n%5E2m%2B112k%5E2nm%5E2%2B98km%5E3 ALIGN=MIDDLE  ALT=\"32k%5E3n%5E2m%2B112k%5E2nm%5E2%2B98km%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > completely factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2km%284kn%2B7m%29%5E2 ALIGN=MIDDLE  ALT=\"2km%284kn%2B7m%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9k%5E2-16m%5E2 ALIGN=MIDDLE  ALT=\"9k%5E2-16m%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=%283k%29%5E2-16m%5E2 ALIGN=MIDDLE  ALT=\"%283k%29%5E2-16m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9k%5E2 ALIGN=MIDDLE  ALT=\"9k%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283k%29%5E2 ALIGN=MIDDLE  ALT=\"%283k%29%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=%283k%29%5E2-%284m%29%5E2 ALIGN=MIDDLE  ALT=\"%283k%29%5E2-%284m%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16m%5E2 ALIGN=MIDDLE  ALT=\"16m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284m%29%5E2 ALIGN=MIDDLE  ALT=\"%284m%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Notice how we have a difference of squares. So let's use the difference of squares formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2-B%5E2=%28A%2BB%29%28A-B%29 ALIGN=MIDDLE  ALT=\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to factor the expression:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283k%2B4m%29%283k-4m%29 ALIGN=MIDDLE  ALT=\"%283k%2B4m%29%283k-4m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the expression using the difference of squares.\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9k%5E2-16m%5E2 ALIGN=MIDDLE  ALT=\"9k%5E2-16m%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283k%2B4m%29%283k-4m%29 ALIGN=MIDDLE  ALT=\"%283k%2B4m%29%283k-4m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In other words <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9k%5E2-16m%5E2=%283k%2B4m%29%283k-4m%29 ALIGN=MIDDLE  ALT=\"9k%5E2-16m%5E2=%283k%2B4m%29%283k-4m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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