document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=182573>Question 182573</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780136028956/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> simplify the rational expression:<BR>\n" );
document.write( "x^3-2x^2+x-2/x-2\r<BR>\n" );
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document.write( "Thank you so much for the help!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #137053 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=137053\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \frac{x^3-2x^2+x-2}{x-2}\">\r<BR>\n" );
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document.write( "Before you do anything else, determine values of the variable that must be excluded.  For a rational expression, set the denominator equal to zero and solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math x - 2 = 0 \ \ \Rightarrow\ \ x = 2\">\r<BR>\n" );
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document.write( "So, regardless of what the expression looks like after simplification, the value 2 must remain excluded.\r<BR>\n" );
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document.write( "Use Polynomial Long Division:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> goes into <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^3,\  x^2\"> times, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{              }\math x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^2 \times x - 2 = x^3 - 2x^2\"> so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{              }\math x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \underline{x^3\  - 2x^2\">\r<BR>\n" );
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document.write( "Change signs of subtrahend and add:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{              }\math x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \underline{x^3\  - 2x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ 0\">\r<BR>\n" );
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document.write( "Bring down the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{              }\math x^2\ \ \ \\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \underline{x^3\  - 2x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> goes into <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0,\  0\"> times, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{              }\math x^2\ \ \ \, 0\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \underline{x^3\  - 2x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0 \times x - 2 = 0\"> so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{              }\math x^2\ \ \ \, 0\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \underline{x^3\  - 2x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}\">\r<BR>\n" );
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document.write( "Change signs of subtrahend and add:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{              }\math x^2\ \ \ \, 0\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \underline{x^3\  - 2x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ x\">\r<BR>\n" );
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document.write( "Bring down the -2:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{              }\math x^2\ \ \ \, 0\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \underline{x^3\  - 2x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ x\ -2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> goes into <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x,\  1\"> times, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{              }\math x^2\ \ \ \, 0\ +1\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \underline{x^3\  - 2x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ x\ -2\">\r<BR>\n" );
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document.write( "Finally:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2 \text{  }\math \overline{|x^3\ -2x^2\ +x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \underline{x^3\  - 2x^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ 0\ \ \,+x\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ \,0}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ x\ -2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \underline{\ x\ -2}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text{       }\math \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,0\">\r<BR>\n" );
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document.write( "Since the remainder is zero, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x - 2\"> divides <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x^3 -2x^2 + x =2\"> evenly.  Therefore we can say:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{          }\math \frac{x^3-2x^2+x-2}{x-2} = x^2 + 1\ \ \Leftrightarrow\ \ x \neq 2\">\r<BR>\n" );
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document.write( "John<BR>\n" );
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