document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=248452>Question 248452</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I just learned this today and not sure how to do this.  Please help by showing how to arrive at the arrive at the answer. I have a quiz tomorrow on it. Thank you!!\r<BR>\n" );
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document.write( "x^4-15x^2y^2-16y^4      </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #181058 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=181058\">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=x%5E4-15x%5E2y%5E2-16y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4-15x%5E2y%5E2-16y%5E4\"   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=x%5E4 ALIGN=MIDDLE  ALT=\"x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16y%5E4 ALIGN=MIDDLE  ALT=\"-16y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and -16 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient -16 to get -16. Now what two numbers multiply to -16 and add to the  middle coefficient -15? Let's list all of the factors of -16:\r<BR>\n" );
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document.write( "Factors of -16:\r<BR>\n" );
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document.write( "1,2,4,8\r<BR>\n" );
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document.write( "-1,-2,-4,-8 ...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 -16\r<BR>\n" );
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document.write( "(1)*(-16)\r<BR>\n" );
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document.write( "(2)*(-8)\r<BR>\n" );
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document.write( "(-1)*(16)\r<BR>\n" );
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document.write( "(-2)*(8)\r<BR>\n" );
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document.write( "note: remember, the product of a negative and a positive number is a negative number\r<BR>\n" );
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document.write( "Now which of these pairs add to -15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -15\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD   align=\"center\"><font color=red>1</font></TD><TD   align=\"center\"><font color=red>-16</font></TD><TD   align=\"center\"><font color=red>1+(-16)=-15</font></TD></tr><tr><TD   align=\"center\"><font color=black>2</font></TD><TD   align=\"center\"><font color=black>-8</font></TD><TD   align=\"center\"><font color=black>2+(-8)=-6</font></TD></tr><tr><TD   align=\"center\"><font color=black>4</font></TD><TD   align=\"center\"><font color=black>-4</font></TD><TD   align=\"center\"><font color=black>4+(-4)=0</font></TD></tr><tr><TD   align=\"center\"><font color=black>-1</font></TD><TD   align=\"center\"><font color=black>16</font></TD><TD   align=\"center\"><font color=black>-1+16=15</font></TD></tr><tr><TD   align=\"center\"><font color=black>-2</font></TD><TD   align=\"center\"><font color=black>8</font></TD><TD   align=\"center\"><font color=black>-2+8=6</font></TD></tr><tr><TD   align=\"center\"><font color=black>-4</font></TD><TD   align=\"center\"><font color=black>4</font></TD><TD   align=\"center\"><font color=black>-4+4=0</font></TD></tr>\n" );
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document.write( "From this list we can see that 1 and -16 add up to -15 and multiply to -16\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=x%5E4-15x%5E2y%5E2-16y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4-15x%5E2y%5E2-16y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15x%5E2y%5E2 ALIGN=MIDDLE  ALT=\"-15x%5E2y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2y%5E2-16x%5E2y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2y%5E2-16x%5E2y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2y%5E2-16x%5E2y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2y%5E2-16x%5E2y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > combines to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15x%5E2y%5E2 ALIGN=MIDDLE  ALT=\"-15x%5E2y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15x%5E2y%5E2 ALIGN=MIDDLE  ALT=\"-15x%5E2y%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=x%5E4%2Bhighlight%28x%5E2y%5E2-16x%5E2y%5E2%29-16y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bhighlight%28x%5E2y%5E2-16x%5E2y%5E2%29-16y%5E4\"   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=x%5E4%2Bx%5E2y%5E2-16x%5E2y%5E2-16y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E2y%5E2-16x%5E2y%5E2-16y%5E4\"   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=%28x%5E4%2Bx%5E2y%5E2%29%2B%28-16x%5E2y%5E2-16y%5E4%29 ALIGN=MIDDLE  ALT=\"%28x%5E4%2Bx%5E2y%5E2%29%2B%28-16x%5E2y%5E2-16y%5E4%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=x%5E2%28x%5E2%2By%5E2%29-16y%5E2%28x%5E2%2By%5E2%29 ALIGN=MIDDLE  ALT=\"x%5E2%28x%5E2%2By%5E2%29-16y%5E2%28x%5E2%2By%5E2%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=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   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=-16y%5E2 ALIGN=MIDDLE  ALT=\"-16y%5E2\"   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=%28x%5E2-16y%5E2%29%28x%5E2%2By%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2-16y%5E2%29%28x%5E2%2By%5E2%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=x%5E2%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2\"   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=x%5E4%2Bx%5E2y%5E2-16x%5E2y%5E2-16y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E2y%5E2-16x%5E2y%5E2-16y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-16y%5E2%29%28x%5E2%2By%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2-16y%5E2%29%28x%5E2%2By%5E2%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=x%5E4-15x%5E2y%5E2-16y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4-15x%5E2y%5E2-16y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-16y%5E2%29%28x%5E2%2By%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2-16y%5E2%29%28x%5E2%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4-15x%5E2y%5E2-16y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4-15x%5E2y%5E2-16y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2Bx%5E2y%5E2-16x%5E2y%5E2-16y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4%2Bx%5E2y%5E2-16x%5E2y%5E2-16y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Finally, the expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-16y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-16y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a difference of squares which means that it factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4y%29%28x-4y%29 ALIGN=MIDDLE  ALT=\"%28x%2B4y%29%28x-4y%29\"   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=x%5E4-15x%5E2y%5E2-16y%5E4 ALIGN=MIDDLE  ALT=\"x%5E4-15x%5E2y%5E2-16y%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > completely factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4y%29%28x-4y%29%28x%5E2%2By%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%2B4y%29%28x-4y%29%28x%5E2%2By%5E2%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=x%5E4-15x%5E2y%5E2-16y%5E4=%28x%2B4y%29%28x-4y%29%28x%5E2%2By%5E2%29 ALIGN=MIDDLE  ALT=\"x%5E4-15x%5E2y%5E2-16y%5E4=%28x%2B4y%29%28x-4y%29%28x%5E2%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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