document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=734948>Question 734948</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to find the slant asymptotes of \r<BR>\n" );
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document.write( "a. x^3+x^2/x^2-4<BR>\n" );
document.write( "b. 2x^2+10x-12/x^2+x-6\r<BR>\n" );
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document.write( "I understand how to find the Vertical and Horizontal, but I'm confused on the slant portion. Please help! thank you!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #449211 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=449211\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> to find the slant asymptotes of\r<BR>\n" );
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document.write( "a. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29 ALIGN=MIDDLE  ALT=\"%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29\"   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%28x%2B1%29%2F%28x%5E2-2%5E2%29 ALIGN=MIDDLE  ALT=\"x%5E2%28x%2B1%29%2F%28x%5E2-2%5E2%29\"   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%28x%2B1%29%2F%28%28x-2%29%28x%2B2%29%29 ALIGN=MIDDLE  ALT=\"x%5E2%28x%2B1%29%2F%28%28x-2%29%28x%2B2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The vertical asymptotes (and any restrictions on the domain) come from the zeroes of the denominator, so I'll 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/plot-formula.mpl?expression=%28%28x-2%29%28x%2B2%29%29=0 ALIGN=MIDDLE  ALT=\"%28%28x-2%29%28x%2B2%29%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2=0 ALIGN=MIDDLE  ALT=\"x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >...=>...<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2=0 ALIGN=MIDDLE  ALT=\"x%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >...=>...<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then the domain is all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-values ALIGN=MIDDLE  ALT=\"x-values\"   DISABLED_style= \"max-width:90%; height:auto;\" > other than ± <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the two vertical asymptotes are at<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2 ALIGN=MIDDLE  ALT=\"x+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since the degree is greater in the denominator than in the numerator, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-values ALIGN=MIDDLE  ALT=\"y-values\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be dragged down to the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-axis ALIGN=MIDDLE  ALT=\"x-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the horizontal asymptote is therefore \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+0 ALIGN=MIDDLE  ALT=\"y+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\"  (the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-axis ALIGN=MIDDLE  ALT=\"x-axis\"   DISABLED_style= \"max-width:90%; height:auto;\" >) .\r<BR>\n" );
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document.write( "The slant asymptote:  use the long division to find it\r<BR>\n" );
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document.write( "----------<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1 ALIGN=MIDDLE  ALT=\"x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------|<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2Bx%5E2 ALIGN=MIDDLE  ALT=\"x%5E3%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4 ALIGN=MIDDLE  ALT=\"x%5E2-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >)..<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-4x ALIGN=MIDDLE  ALT=\"x%5E3-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "_________________<BR>\n" );
document.write( "----------<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2B4x%2Bx%5E2 ALIGN=MIDDLE  ALT=\"0%2B4x%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------------ <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4 ALIGN=MIDDLE  ALT=\"x%5E2-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "________________<BR>\n" );
document.write( "---------------<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B4 ALIGN=MIDDLE  ALT=\"4x%2B4\"   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=4%28x%2B1%29 ALIGN=MIDDLE  ALT=\"4%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1 ALIGN=MIDDLE  ALT=\"x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of the numerator\r<BR>\n" );
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document.write( "the slant asymptote:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x+%2B+1 ALIGN=MIDDLE  ALT=\"y+=+x+%2B+1\"   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=drawing%28600%2C600%2C+-10%2C10%2C+-10%2C+10%2C+blue%28line%282%2C10%2C2%2C-10%29%29%2C++blue%28line%28-2%2C10%2C-2%2C-10%29%29%2C+graph%28+600%2C600%2C+-10%2C10%2C+-10%2C+10%2C+%28x%5E3%2Bx%5E2%29%2F%28x%5E2-4%29%2Cx%2B1%29%2C0%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "b. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E2%2B10x-12%29%2F%28x%5E2%2Bx-6%29 ALIGN=MIDDLE  ALT=\"%282x%5E2%2B10x-12%29%2F%28x%5E2%2Bx-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "horizontal asymptote: -> to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12%2F-6=2 ALIGN=MIDDLE  ALT=\"-12%2F-6=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so,horizontal asymptote is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "continue factoring\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E2%2B10x-12%29%2F%28x%5E2%2Bx-6%29+ ALIGN=MIDDLE  ALT=\"%282x%5E2%2B10x-12%29%2F%28x%5E2%2Bx-6%29+\"   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=2%28x%5E2%2B5x-6%29%2F%28x%5E2%2Bx-6%29+ ALIGN=MIDDLE  ALT=\"2%28x%5E2%2B5x-6%29%2F%28x%5E2%2Bx-6%29+\"   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=2%28x%5E2-x%2B6x-6%29%2F%28x%5E2%2B3x-2x-6%29+ ALIGN=MIDDLE  ALT=\"2%28x%5E2-x%2B6x-6%29%2F%28x%5E2%2B3x-2x-6%29+\"   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=2%28%28x%5E2-x%29%2B%286x-6%29%29%2F%28%28x%5E2%2B3x%29-%282x%2B6%29%29+ ALIGN=MIDDLE  ALT=\"2%28%28x%5E2-x%29%2B%286x-6%29%29%2F%28%28x%5E2%2B3x%29-%282x%2B6%29%29+\"   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=2%28x%28x-1%29%2B6%28x-1%29%29%2F%28x%28x%2B3%29-2%28x%2B3%29%29+ ALIGN=MIDDLE  ALT=\"2%28x%28x-1%29%2B6%28x-1%29%29%2F%28x%28x%2B3%29-2%28x%2B3%29%29+\"   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=2%28%28x%2B6%29%28x-1%29%29%2F%28%28x-2%29%28x%2B3%29%29+ ALIGN=MIDDLE  ALT=\"2%28%28x%2B6%29%28x-1%29%29%2F%28%28x-2%29%28x%2B3%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "find values that make denominator equal to zero:\r<BR>\n" );
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document.write( "if<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2=0+ ALIGN=MIDDLE  ALT=\"x-2=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >....->...<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3=0+ ALIGN=MIDDLE  ALT=\"x%2B3=0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >....->...<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "these are vertical asymptotes\r<BR>\n" );
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document.write( "-there is no slant asymptote in this case\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C600%2C+-10%2C10%2C+-10%2C+10%2C+blue%28line%282%2C10%2C2%2C-10%29%29%2C++blue%28line%28-3%2C10%2C-3%2C-10%29%29%2C+graph%28+600%2C600%2C+-10%2C10%2C+-10%2C+10%2C+%282x%5E2%2B10x-12%29%2F%28x%5E2%2Bx-6%29%2C2%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " </SPAN>\n" );
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