document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163607>Question 1163607</A>: <I><SPAN STYLE=\"word-break: break-all;\"> oblique asymptote:<BR>\n" );
document.write( "If a given rational function of higher degree (say 6) does not have a horizontal asymptote, to check for the oblique asymptote...<BR>\n" );
document.write( "   ...Do you just keep doing polymomial division repeatedly till you come down to the quotient that is of form mx+b ?<BR>\n" );
document.write( "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 #787743 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=787743\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There is an oblique (linear) asymptote only if the degree of the numerator is 1 more than the degree of the denominator.<br><BR>\n" );
document.write( "So there is never any need to \"keep doing polynomial division\" to find an asymptote.  If there is an oblique asymptote, a single division will find it.<br><BR>\n" );
document.write( "Example 1: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B2x%29%2F%28x%2B1%29+=+%28x%2B1%29-1%2F%28x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B2x%29%2F%28x%2B1%29+=+%28x%2B1%29-1%2F%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph of the rational function and the oblique asymptote, y=x+1:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-10%2C10%2C%28x%5E2%2B2x%29%2F%28x%2B1%29%2Cx%2B1%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-5%2C5%2C-10%2C10%2C%28x%5E2%2B2x%29%2F%28x%2B1%29%2Cx%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Example 2: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3%2B3x%5E2%2B3x%29%2F%28x%5E2%2B2x%2B1%29+=+%28x%2B1%29-1%2F%28x%5E2%2B2x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%5E3%2B3x%5E2%2B3x%29%2F%28x%5E2%2B2x%2B1%29+=+%28x%2B1%29-1%2F%28x%5E2%2B2x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph of the rational function and the oblique asymptote, y=x+1:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-20%2C20%2C%28x%5E3%2B3x%5E2%2B3x%29%2F%28x%5E2%2B2x%2B1%29%2Cx%2B1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "If the degree of the numerator is more than 1 greater than the degree of the denominator, then you get an asymptote which is not oblique (linear).<br><BR>\n" );
document.write( "For example, if the degree of the numerator is 2 more than the degree of the denominator, the asymptote is degree 2 (quadratic -- a parabola).<br><BR>\n" );
document.write( "Example 3: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3%2B3x%5E2%2B3x%29%2F%28x%2B1%29+=+%28x%5E2%2B2x%2B1%29-1%2F%28x%2B1%29 ALIGN=MIDDLE  ALT=\"%28x%5E3%2B3x%5E2%2B3x%29%2F%28x%2B1%29+=+%28x%5E2%2B2x%2B1%29-1%2F%28x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph of the rational function and the quadratic asymptote:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-20%2C20%2C%28x%5E3%2B3x%5E2%2B3x%29%2F%28x%2B1%29%2Cx%5E2%2B2x%2B1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In general, if the degree of the numerator is n more than the degree of the denominator, the asymptote will be of degree n.<br><BR>\n" );
document.write( "Example 4: numerator degree 4, denominator degree 1; asymptote degree 3: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E4%2B4x%5E3%2B6x%5E2%2B4x%29%2F%28x%2B1%29+=+%28x%5E3%2B3x%5E2%2B3x%2B1%29-1%2F%28x%2B1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-20%2C20%2C%28x%5E4%2B4x%5E3%2B6x%5E2%2B4x%29%2F%28x%2B1%29%2C%28x%5E3%2B3x%5E2%2B3x%2B1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );