document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=131936>Question 131936</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factoring Polynomials -\r<BR>\n" );
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document.write( "Please factor the following:\r<BR>\n" );
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document.write( "w4 - 8w2 - 9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96317 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=96317\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I presume you meant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E4+-+8w%5E2+-+9 ALIGN=MIDDLE  ALT=\"w%5E4+-+8w%5E2+-+9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=w%5E2 ALIGN=MIDDLE  ALT=\"x=w%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then substitute:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x-9 ALIGN=MIDDLE  ALT=\"x%5E2-8x-9\"   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=-9%2A1=-9 ALIGN=MIDDLE  ALT=\"-9%2A1=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9%2B1=-8 ALIGN=MIDDLE  ALT=\"-9%2B1=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x-9 ALIGN=MIDDLE  ALT=\"x%5E2-8x-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-9%29%28x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x-9%29%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=w%5E2 ALIGN=MIDDLE  ALT=\"x=w%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so two of the factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E4+-+8w%5E2+-+9 ALIGN=MIDDLE  ALT=\"w%5E4+-+8w%5E2+-+9\"   DISABLED_style= \"max-width:90%; height:auto;\" > are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28w%5E2-9%29%28w%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"%28w%5E2-9%29%28w%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The first of these factors is the difference of two squares and factors directly to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28w%5E2-9%29=%28w-3%29%28w%2B3%29 ALIGN=MIDDLE  ALT=\"%28w%5E2-9%29=%28w-3%29%28w%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For the other factor, we need to use the fact that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a zero of a single variable polynomial in x if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-a ALIGN=MIDDLE  ALT=\"x-a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of the polynomial.\r<BR>\n" );
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document.write( "So, solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E2%2B1=0 ALIGN=MIDDLE  ALT=\"w%5E2%2B1=0\"   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=w%5E2=-1 ALIGN=MIDDLE  ALT=\"w%5E2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+i ALIGN=MIDDLE  ALT=\"w+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+-i ALIGN=MIDDLE  ALT=\"w+=+-i\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2=-1 ALIGN=MIDDLE  ALT=\"i%5E2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, therefore the two factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E2%2B1 ALIGN=MIDDLE  ALT=\"w%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28w-i%29%28w%2Bi%29 ALIGN=MIDDLE  ALT=\"%28w-i%29%28w%2Bi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In summary:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E4+-+8w%5E2+-+9=%28w-3%29%28w%2B3%29%28w-i%29%28w%2Bi%29 ALIGN=MIDDLE  ALT=\"w%5E4+-+8w%5E2+-+9=%28w-3%29%28w%2B3%29%28w-i%29%28w%2Bi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If you were only supposed to factor over the real numbers, then your answer would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E4+-+8w%5E2+-+9=%28w-3%29%28w%2B3%29%28w%5E2%2B1%29 ALIGN=MIDDLE  ALT=\"w%5E4+-+8w%5E2+-+9=%28w-3%29%28w%2B3%29%28w%5E2%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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