document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=113821>Question 113821</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 6.	Determine the behavior of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Domain :	<BR>\n" );
document.write( "x – intercept(s) :<BR>\n" );
document.write( "Hole(s) :	<BR>\n" );
document.write( "y – intercept :<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82830 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=82830\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <b>Domain:</b>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=0 ALIGN=MIDDLE  ALT=\"x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the \r<BR>\n" );
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document.write( "denominator zero, then we must exclude them from the domain.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0%2B3 ALIGN=MIDDLE  ALT=\"x=0%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >Add 3 to both sides\r<BR>\n" );
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document.write( "<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;\" > Combine like terms on the right side\r<BR>\n" );
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document.write( "Since <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;\" > makes the denominator equal to zero, this means we must exclude <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;\" > from our domain\r<BR>\n" );
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document.write( "So our domain is:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \textrm{\left{x|x\in\mathbb{R} x\neq3\right}}\">\r<BR>\n" );
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document.write( "which in plain English reads: x is the set of all real numbers except <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E3 ALIGN=MIDDLE  ALT=\"x%3C%3E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So our domain looks like this in interval notation\r<BR>\n" );
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document.write( "<b>x-intercept(s):</b>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=%28x%5E2%2Bx-6%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"0=%28x%5E2%2Bx-6%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > To find the x-intercepts, set f(x) (which is y) equal to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx-6=0 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx-6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > This means the numerator is equal to zero. Remember the denominator cannot equal zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%28x-2%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%28x-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side (note: if you need help with factoring, check out this <BR>\n" );
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document.write( "Now set each factor equal to zero:\r<BR>\n" );
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document.write( "<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;\" > or <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;\" >  Now solve for x in each case\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So our solutions are <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;\" > or <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( "This means the x-intercepts are (-3,0) and (2,0)\r<BR>\n" );
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document.write( "<b>Hole(s):</b>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%2B3%29%28x-2%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%2B3%29%28x-2%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor\r<BR>\n" );
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document.write( "Since the expression does not simplify further, there are no exceptions to make about the two expressions. So in this case there are no holes in this function\r<BR>\n" );
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document.write( "<b>y-intercept(s):</b>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=%280%5E2%2B0-6%29%2F%280-3%29 ALIGN=MIDDLE  ALT=\"f%280%29=%280%5E2%2B0-6%29%2F%280-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > To find the y-intercept, plug in x=0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=%28-6%29%2F%28-3%29 ALIGN=MIDDLE  ALT=\"f%280%29=%28-6%29%2F%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=2 ALIGN=MIDDLE  ALT=\"f%280%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So the y intercept is (0,2)\r<BR>\n" );
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document.write( "<b>Vertical Asymptote(s):</b>\r<BR>\n" );
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document.write( "Since the value x=3 is excluded from the domain, and there are no holes, there is one vertical asymptote at x=3\r<BR>\n" );
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document.write( "So the vertical asymptote is <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( "<b>Oblique Asymptote:</b>\r<BR>\n" );
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document.write( "To find the oblique asymptote, divide <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2Bx-6%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%2Bx-6%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > using synthetic division\r<BR>\n" );
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document.write( "Start with the given expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2+%2B+x+-+6%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"%28x%5E2+%2B+x+-+6%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "First lets find our test zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-3=0 ALIGN=MIDDLE  ALT=\"x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator <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;\" > equal to zero\r<BR>\n" );
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document.write( "<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;\" > Solve for x.\r<BR>\n" );
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document.write( "so our test zero is 3\r<BR>\n" );
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document.write( "Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >1</TD><TD >-6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >1</TD><TD >-6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 3 by 1 and place the product (which is 3)  right underneath the second  coefficient (which is 1)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >1</TD><TD >-6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 3 and 1 to get 4. Place the sum right underneath 3.\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >1</TD><TD >-6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >4</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 3 by 4 and place the product (which is 12)  right underneath the third  coefficient (which is -6)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >1</TD><TD >-6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >12</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >4</TD><TD ></TD></TR>\n" );
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document.write( "    Add 12 and -6 to get 6. Place the sum right underneath 12.\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >1</TD><TD >-6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >12</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >4</TD><TD >6</TD></TR>\n" );
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document.write( "Since the last column adds to 6, we have a remainder of 6. This means <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;\" > is <b>not</b> a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+x+-+6 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+x+-+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets look at the bottom row of coefficients:\r<BR>\n" );
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document.write( "The first 2 coefficients (1,4) form the quotient\r<BR>\n" );
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document.write( "So the oblique asymptote is the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%2B4 ALIGN=MIDDLE  ALT=\"y=x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice if we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2%2Bx-6%29%2F%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and it's asymptotes, we can visually verify our answers:\r<BR>\n" );
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