document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086481>Question 1086481</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If f(x)=(x^5-6x^4+7x^3+6x^2-8x)/(x^2-2x)\r<BR>\n" );
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document.write( "(a) Find all intercept(s) if any.\r<BR>\n" );
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document.write( "(b) Find all asymptote(s), hole(s), gap(s),jump(s) if any.\r<BR>\n" );
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document.write( "(c) Graph.\r<BR>\n" );
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document.write( "(d) Give the domain and range. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #700715 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=700715\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x)=(x^5-6x^4+7x^3+6x^2-8x)/(x^2-2x) <BR>\n" );
document.write( "factor out an x(x^4-6x^3+7x^2+6x-8)/x(x-2)<BR>\n" );
document.write( "x cancel,<BR>\n" );
document.write( "x can't be 0, but they cancel, so there is a hole at x=0.<BR>\n" );
document.write( "suspect 2 is a root, so (x-2) a factor<BR>\n" );
document.write( "2/1===-6===7===6===-8<BR>\n" );
document.write( "=1====-4=-1====4===0<BR>\n" );
document.write( "it is<BR>\n" );
document.write( "the other factor is (x^3-4x^2-x+4)<BR>\n" );
document.write( "if x=1, that will be a root by inspection (just put 1 in for x and see if the coefficients add up to 0.  They do)<BR>\n" );
document.write( "1/1====-4===-1====4<BR>\n" );
document.write( "==1====-3===-4===0<BR>\n" );
document.write( "(x-1) is a root.<BR>\n" );
document.write( "That leaves x^2-3x-4, and that factors into (x-4)(x+1)<BR>\n" );
document.write( "The roots are -1,0,1,2,4.  But the denominator factors into x(x-2), so 0 and 2 are holes and -1,1,4 are roots.<BR>\n" );
document.write( "Factored, the function is<BR>\n" );
document.write( "(x+1)(x-1)(x-2)(x-4)*x/x(x-2)<BR>\n" );
document.write( "That becomes (x+1)(x-1)(x-4).<BR>\n" );
document.write( "Domain is all x except x=0,2<BR>\n" );
document.write( "range is all real numbers<BR>\n" );
document.write( "Polynomial division gives a quotient of x^3-4x^2-4x+4, which has no remainder.  The original function has holes at x=0 and x=2; the slant asymptote does not.<BR>\n" );
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