document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=790305>Question 790305</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the vertical, horizontal, and slant asymptotes, if any, for f(x)=5x^3+49x^2+127x+75/x^2+9x+18 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #479111 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=479111\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You meant<BR>\n" );
document.write( "f(x)=(5x^3+49x^2+127x+75)/(x^2+9x+18)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%5E3%2B49x%5E2%2B127x%2B75%29%2F%28x%5E2%2B9x%2B18%29 ALIGN=MIDDLE  ALT=\"%285x%5E3%2B49x%5E2%2B127x%2B75%29%2F%28x%5E2%2B9x%2B18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Factoring, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B9x%2B18=%28x%2B6%29%28x%2B3%29 ALIGN=MIDDLE  ALT=\"x%5E2%2B9x%2B18=%28x%2B6%29%28x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3%2B49x%5E2%2B127x%2B75 ALIGN=MIDDLE  ALT=\"5x%5E3%2B49x%5E2%2B127x%2B75\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3 ALIGN=MIDDLE  ALT=\"x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3%2B49x%5E2%2B127x%2B75=%285x%5E2%2B34x%2B25%29%28x%2B3%29 ALIGN=MIDDLE  ALT=\"5x%5E3%2B49x%5E2%2B127x%2B75=%285x%5E2%2B34x%2B25%29%28x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%285x%5E3%2B49x%5E2%2B127x%2B75%29%2F%28x%5E2%2B9x%2B18%29=%285x%5E2%2B34x%2B25%29%28x%2B3%29%2F%28%28x%2B6%29%28x%2B3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-3%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-6%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are vertical asymptotes,<BR>\n" );
document.write( "and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E-3 ALIGN=MIDDLE  ALT=\"x%3C%3E-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%285x%5E2%2B34x%2B25%29%2F%28x%2B6%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%285x%5E2%2B34x%2B25%29%2F%28x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "If we divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B34x%2B25 ALIGN=MIDDLE  ALT=\"5x%5E2%2B34x%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B6 ALIGN=MIDDLE  ALT=\"x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "the quotient is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B4 ALIGN=MIDDLE  ALT=\"5x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the remainder is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "In other words <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E2%2B34x%2B25=%28x%2B6%29%285x%2B4%29%2B1 ALIGN=MIDDLE  ALT=\"5x%5E2%2B34x%2B25=%28x%2B6%29%285x%2B4%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C-6 ALIGN=MIDDLE  ALT=\"x%3C-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%285x%5E2%2B34x%2B25%29%2F%28x%2B6%29=%28%28x%2B6%29%285x%2B4%29%2B1%29%2F%28x%2B6%29=%28x%2B6%29%285x%2B4%29%2F%28x%2B6%29%2B1%2F%28x%2B6%29=5x%2B4%2B1%2F%28x%2B6%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5x%2B4 ALIGN=MIDDLE  ALT=\"y=5x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%29 ALIGN=MIDDLE  ALT=\"abs%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > increases.<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=5x%2B4%29 ALIGN=MIDDLE  ALT=\"highlight%28y=5x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slant asymptote on both sides.<BR>\n" );
document.write( " <BR>\n" );
document.write( "IMPORTANT NOTE:<BR>\n" );
document.write( "Those parentheses are essential if you cannot type the equation as the \"two-story\" version,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%285x%5E3%2B49x%5E2%2B127x%2B75%29%2F%28x%5E2%2B9x%2B18%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%285x%5E3%2B49x%5E2%2B127x%2B75%29%2F%28x%5E2%2B9x%2B18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with a long horizontal line in the middle.<BR>\n" );
document.write( "The horizontal line implies the parentheses.<BR>\n" );
document.write( "Without the parentheses, you get<BR>\n" );
document.write( "f(x)=5x^3+49x^2+127x+75/x^2+9x+18=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%5E3%2B49x%5E2%2B127x%2B75%2Fx%5E2%2B9x%2B18 ALIGN=MIDDLE  ALT=\"5x%5E3%2B49x%5E2%2B127x%2B75%2Fx%5E2%2B9x%2B18\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );