document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=132850>Question 132850</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Rewrite the expression x^3-2x^2y+xy^2 so that it is easy to calculate the its value without a calculator when x=21 and y=19. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #97074 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=97074\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If you factor this expression, then it will be much easier to calculate the result\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2y%2Bxy%5E2 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2y%2Bxy%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%5E2-2xy%2By%5E2%29 ALIGN=MIDDLE  ALT=\"x%28x%5E2-2xy%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the GCF <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's focus on the inner expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-2xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-2xy%2By%5E2\"   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=1x%5E2 ALIGN=MIDDLE  ALT=\"1x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the last term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the coefficients are 1 and 1 respectively.\r<BR>\n" );
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document.write( "Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the  middle coefficient -2? Let's list all of the factors of 1:\r<BR>\n" );
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document.write( "Factors of 1:\r<BR>\n" );
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document.write( "1\r<BR>\n" );
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document.write( "-1 ...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 1\r<BR>\n" );
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document.write( "1*1\r<BR>\n" );
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document.write( "(-1)*(-1)\r<BR>\n" );
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document.write( "note: remember two negative numbers multiplied together make a positive number\r<BR>\n" );
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document.write( "Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2\r<BR>\n" );
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document.write( "<th>First Number</th><th>Second Number</th><th>Sum</th><tr><TD  align=\"center\">1</TD><TD  align=\"center\">1</TD><TD >1+1=2</TD></tr><tr><TD  align=\"center\">-1</TD><TD  align=\"center\">-1</TD><TD >-1+(-1)=-2</TD></tr>\n" );
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document.write( "From this list we can see that -1 and -1 add up to -2 and multiply to 1\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%5E2-2xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-2xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2xy ALIGN=MIDDLE  ALT=\"-2xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-xy-xy ALIGN=MIDDLE  ALT=\"-xy-xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-xy-xy ALIGN=MIDDLE  ALT=\"-xy-xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2xy ALIGN=MIDDLE  ALT=\"-2xy\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So it is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2xy ALIGN=MIDDLE  ALT=\"-2xy\"   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%5E2%2Bhighlight%28-xy-xy%29%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bhighlight%28-xy-xy%29%2By%5E2\"   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%5E2-xy-xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-xy-xy%2By%5E2\"   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%5E2-xy%29%2B%28-xy%2By%5E2%29 ALIGN=MIDDLE  ALT=\"%28x%5E2-xy%29%2B%28-xy%2By%5E2%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%28x-y%29-y%28x-y%29 ALIGN=MIDDLE  ALT=\"x%28x-y%29-y%28x-y%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 ALIGN=MIDDLE  ALT=\"x\"   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=-y ALIGN=MIDDLE  ALT=\"-y\"   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-y%29%28x-y%29 ALIGN=MIDDLE  ALT=\"%28x-y%29%28x-y%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-y ALIGN=MIDDLE  ALT=\"x-y\"   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%5E2-xy-xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-xy-xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-y%29%28x-y%29 ALIGN=MIDDLE  ALT=\"%28x-y%29%28x-y%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%5E2-2xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-2xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-y%29%28x-y%29 ALIGN=MIDDLE  ALT=\"%28x-y%29%28x-y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-2xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-xy-xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-xy-xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "note:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-y%29%28x-y%29 ALIGN=MIDDLE  ALT=\"%28x-y%29%28x-y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is equivalent to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-y%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > since the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y ALIGN=MIDDLE  ALT=\"x-y\"   DISABLED_style= \"max-width:90%; height:auto;\" > occurs twice. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-2xy%2By%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-2xy%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > also factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-y%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our expression goes from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x%5E2-2xy%2By%5E2%29 ALIGN=MIDDLE  ALT=\"x%28x%5E2-2xy%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and factors further to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-y%29%5E2 ALIGN=MIDDLE  ALT=\"x%28x-y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-2x%5E2y%2Bxy%5E2 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2y%2Bxy%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x-y%29%5E2 ALIGN=MIDDLE  ALT=\"x%28x-y%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now let's evaluate the expression when  x=21 and y=19\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%2821-19%29%5E2 ALIGN=MIDDLE  ALT=\"21%2821-19%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now plug in x=21 and y=19\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%282%29%5E2 ALIGN=MIDDLE  ALT=\"21%282%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%284%29 ALIGN=MIDDLE  ALT=\"21%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 2 to get 4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=84 ALIGN=MIDDLE  ALT=\"84\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply \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%5E3-2x%5E2y%2Bxy%5E2=84 ALIGN=MIDDLE  ALT=\"x%5E3-2x%5E2y%2Bxy%5E2=84\"   DISABLED_style= \"max-width:90%; height:auto;\" > when x=21 and y=19 </SPAN>\n" );
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