document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=232374>Question 232374</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Factor this problem completely\r<BR>\n" );
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document.write( "x^8-1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #171744 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=171744\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are many factoring techniques. But this problem requires only one of them: the \"Difference of squares\" pattern, <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;\" >.<br><BR>\n" );
document.write( "We just have to realize what makes a term a perfect square. Numbers which are perfect squares are not hard to recognize. But variable factors which are perfect squares are not as obvious. For example, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E8 ALIGN=MIDDLE  ALT=\"x%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > would not seem to be a perfect square because 8 is not a perfect square. But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E8 ALIGN=MIDDLE  ALT=\"x%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a perfect square. In fact any variable with an even exponent is a perfect square! The reason for this is the rule for exponents: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E%28x%2Ay%29+=+%28a%5Ex%29%5Ey ALIGN=MIDDLE  ALT=\"a%5E%28x%2Ay%29+=+%28a%5Ex%29%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >. A specific case of this rule is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E%28y%2A2%29+=+%28a%5Ey%292 ALIGN=MIDDLE  ALT=\"a%5E%28y%2A2%29+=+%28a%5Ey%292\"   DISABLED_style= \"max-width:90%; height:auto;\" >. On the left side we have an even exponent (since 2 times a whole number is even). On the right side we have something squared. (Similar logic tells us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E15 ALIGN=MIDDLE  ALT=\"x%5E15\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a perfect cube because the exponent is divisible by 3, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E20 ALIGN=MIDDLE  ALT=\"x%5E20\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a perfect 4th power because the exponent is divisible by 4, etc.)<br><BR>\n" );
document.write( "Now we're ready to factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E8+-+1 ALIGN=MIDDLE  ALT=\"x%5E8+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > as s difference of squares:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E8+-+1+=+%28x%5E4%29%5E2+-+1%5E2+=+%28x%5E4%2B1%29%28x%5E4-1%29 ALIGN=MIDDLE  ALT=\"x%5E8+-+1+=+%28x%5E4%29%5E2+-+1%5E2+=+%28x%5E4%2B1%29%28x%5E4-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When you reduce fractions, you always keep going until you can't go any further. Factoring is like reducing fractions because you keep going until you can't go any further. And if we look at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E4%2B1%29%28x%5E4-1%29 ALIGN=MIDDLE  ALT=\"%28x%5E4%2B1%29%28x%5E4-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and if we have learned what makes terms perfect squares, we'll realize that the second factor is another difference of squares:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E8+-+1+=+%28x%5E4%29%5E2+-+1%5E2+=+%28x%5E4%2B1%29%28x%5E4-1%29+=+%28x%5E4%2B1%29%28x%5E2%2B1%29%28x%5E2-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "... and yet another difference of squares!!<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E8+-+1+=+%28x%5E4%29%5E2+-+1%5E2+=+%28x%5E4%2B1%29%28x%5E4-1%29+=+%28x%5E4%2B1%29%28x%5E2%2B1%29%28x%5E2-1%29++=+%28x%5E4%2B1%29%28x%5E2%2B1%29%28x%2B1%29%28x-1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "... and finally we're done. (Note: There is no \"Sum of squares\" pattern so those factors will not factor further.) </SPAN>\n" );
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