document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=414107>Question 414107</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Why is it impossible to factor out the sum of two squares? I know how to factor the difference but I was just curious. Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #290458 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=290458\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Say you have two variables, x and y. Now I'm going to further say that x and y have nothing in common (except 1, but we're going to ignore that). So we can't factor the GCF out of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By ALIGN=MIDDLE  ALT=\"x%2By\"   DISABLED_style= \"max-width:90%; height:auto;\" > because it would be trivial.\r<BR>\n" );
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document.write( "Now square each term to get <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;\" >. Let's assume that we can factor this.\r<BR>\n" );
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document.write( "So using the AC method, we can think 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;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B0xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B0xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we can clearly see that the 3 coefficients are 1,0, and 1\r<BR>\n" );
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document.write( "Multiply the first and last coefficient to get 1*1 = 1. \r<BR>\n" );
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document.write( "Now the big question is: what two numbers BOTH multiply to 1 (the product above) AND add to 0?\r<BR>\n" );
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document.write( "Any ideas?\r<BR>\n" );
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document.write( "If you're stumped, you have a very good reason as there are no two numbers that fit both conditions. Notice how the two numbers multiply to a positive result. So the two numbers are\r<BR>\n" );
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document.write( "a) both positive, or\r<BR>\n" );
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document.write( "b) both negative\r<BR>\n" );
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document.write( "But if you add two positive numbers (no matter how small), you will NOT get zero. Likewise, adding two negative numbers will NEVER give you zero. In order for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By=0 ALIGN=MIDDLE  ALT=\"x%2By=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be true (where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E0 ALIGN=MIDDLE  ALT=\"x%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C%3E0 ALIGN=MIDDLE  ALT=\"y%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) x and y MUST have different signs. It's not possible otherwise.\r<BR>\n" );
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document.write( "So this shows us that no two numbers both multiply to 1 and add to 0\r<BR>\n" );
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document.write( "This consequently means that <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;\" > cannot be factored.\r<BR>\n" );
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document.write( "If you need more help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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document.write( "Jim </SPAN>\n" );
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