document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=101511>Question 101511</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0534493947/algebrahomeworkh>Intermediate Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Can someone please help me factor this problem completely?<BR>\n" );
document.write( "x^10y^3 - 4x^9y^2 - 21x^8y?????\r<BR>\n" );
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document.write( "Here is what I have as my attempt, but am not sure I am even correct with what I have..<BR>\n" );
document.write( "Simplify:<BR>\n" );
document.write( "x^10y^3 - 36xy^2 - 168xy<BR>\n" );
document.write( "x^10y^3 - 72xy - 168xy\r<BR>\n" );
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document.write( "Combine like terms:<BR>\n" );
document.write( "72xy + 168xy = 240xy\r<BR>\n" );
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document.write( "240xy* x^10*y^3=?\r<BR>\n" );
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document.write( "This problem was given to me by my college Professor, but it is not from our textbook. Thanks. Below are additional directions from Professor:\r<BR>\n" );
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document.write( "**Make sure to superscript all exponents.  If you believe that a polynomial is not factorable, label it as prime.  Show all stages of factoring separately (i.e., show the result of factoring out a GCF before taking the resulting polynomial and factoring it again).**\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #73909 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=73909\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E10y%5E3+-+4x%5E9y%5E2+-+21x%5E8y ALIGN=MIDDLE  ALT=\"x%5E10y%5E3+-+4x%5E9y%5E2+-+21x%5E8y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "First look at the numbers that multiply each of the terms to see if there is are common<BR>\n" );
document.write( "factors. The first term is multiplied by 1 (understood), the second term by -4, and the<BR>\n" );
document.write( "third term by -21.  These three numbers have no common factors, so we don't change them.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next look at the x part of each of the terms. The first term contains <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E10 ALIGN=MIDDLE  ALT=\"x%5E10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the second<BR>\n" );
document.write( "term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E9 ALIGN=MIDDLE  ALT=\"x%5E9\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the third term <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;\" >. <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 common to each term because<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E10+=+x%5E2%2Ax%5E8 ALIGN=MIDDLE  ALT=\"x%5E10+=+x%5E2%2Ax%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E9+=+x%2Ax%5E8 ALIGN=MIDDLE  ALT=\"x%5E9+=+x%2Ax%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we can pull an <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;\" > from each term and<BR>\n" );
document.write( "we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E10y%5E3+-+4x%5E9y%5E2+-+21x%5E8y+=+%28x%5E8%29%2A%28x%5E2%2Ay%5E3+-+4x%2Ay%5E2+-21y%29 ALIGN=MIDDLE  ALT=\"x%5E10y%5E3+-+4x%5E9y%5E2+-+21x%5E8y+=+%28x%5E8%29%2A%28x%5E2%2Ay%5E3+-+4x%2Ay%5E2+-21y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next notice that you have a \"y\" common to all terms. So pull a y out as a multiplier and <BR>\n" );
document.write( "you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E8%29%2A%28x%5E2%2Ay%5E3+-+4x%2Ay%5E2+-21y%29+=+%28x%5E8%29%2A%28y%29%28x%5E2y%5E2+-4xy+-+21%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Notice that we can multiply out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E8%29%2A%28y%29%28x%5E2y%5E2+-4xy-21%29 ALIGN=MIDDLE  ALT=\"%28x%5E8%29%2A%28y%29%28x%5E2y%5E2+-4xy-21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the result should be<BR>\n" );
document.write( "the original given expression. This is just a \"check\".<BR>\n" );
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document.write( "So we are down to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E8%29%2A%28y%29%28x%5E2y%5E2+-4xy-21%29 ALIGN=MIDDLE  ALT=\"%28x%5E8%29%2A%28y%29%28x%5E2y%5E2+-4xy-21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We now need to see if we can factor the<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2y%5E2+-+4xy+-21 ALIGN=MIDDLE  ALT=\"x%5E2y%5E2+-+4xy+-21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Yes we can.  It might be a little easier to see if we wrote <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2y%5E2+ ALIGN=MIDDLE  ALT=\"x%5E2y%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > in an equivalent form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xy%29%5E2 ALIGN=MIDDLE  ALT=\"%28xy%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to change the expression to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xy%29%5E2+-+4xy+-+21 ALIGN=MIDDLE  ALT=\"%28xy%29%5E2+-+4xy+-+21\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we might see it a little more clearly if we let xy = A to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2+-+4A+-+21 ALIGN=MIDDLE  ALT=\"A%5E2+-+4A+-+21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "We can try to factor this into (A + ___)*(A + ___). The underscores have to be two numbers<BR>\n" );
document.write( "that are factors of -21. These numbers could be 21 and 1 or 7 and 3. With their appropriate signs<BR>\n" );
document.write( "they must sum to -4.  -7 and + 3 do that. If they are multiplied they give -21 and if they are<BR>\n" );
document.write( "added they give -4 which is the multiplier of the center term that contains just A.<BR>\n" );
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document.write( "So we can write that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2+-+4A+-+21 ALIGN=MIDDLE  ALT=\"A%5E2+-+4A+-+21\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A+-+7%29%2A%28A%2B+3%29 ALIGN=MIDDLE  ALT=\"%28A+-+7%29%2A%28A%2B+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. But also recall<BR>\n" );
document.write( "that we said A was equal to xy. So now we can substitute xy into the two factors and get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A-7%29%2A%28A%2B3%29+=+%28xy+-+7%29%2A%28xy+%2B+3%29 ALIGN=MIDDLE  ALT=\"%28A-7%29%2A%28A%2B3%29+=+%28xy+-+7%29%2A%28xy+%2B+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "And so we can say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2y%5E2+-+4xy+-+21 ALIGN=MIDDLE  ALT=\"x%5E2y%5E2+-+4xy+-+21\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xy+-+7%29%2A%28xy%2B3%29 ALIGN=MIDDLE  ALT=\"%28xy+-+7%29%2A%28xy%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we can <BR>\n" );
document.write( "carry this result back into this equation that we got earlier:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E8%29%2A%28y%29%28x%5E2y%5E2+-4xy-21%29 ALIGN=MIDDLE  ALT=\"%28x%5E8%29%2A%28y%29%28x%5E2y%5E2+-4xy-21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "In this equation, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2y%5E2+-4xy-21%29 ALIGN=MIDDLE  ALT=\"%28x%5E2y%5E2+-4xy-21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xy+-+7%29%2A%28xy%2B3%29 ALIGN=MIDDLE  ALT=\"%28xy+-+7%29%2A%28xy%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the result is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E8%29%2A%28y%29%2A%28xy+-+7%29%2A%28xy%2B3%29 ALIGN=MIDDLE  ALT=\"%28x%5E8%29%2A%28y%29%2A%28xy+-+7%29%2A%28xy%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and that's the answer to your problem.<BR>\n" );
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document.write( "I know this is confusing, but I hope that you can see your way through it. If you can<BR>\n" );
document.write( "understand all the maneuvering in this you will have made a lot of progress. </SPAN>\n" );
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