document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=389522>Question 389522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the function f(x)=(x^4-625)/((x^2-25)(x+5)(x^2+25))\r<BR>\n" );
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document.write( "a) How do you find the holes in a function?\r<BR>\n" );
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document.write( "b) Identify any vertical asymptotes\r<BR>\n" );
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document.write( "c) Identify any holes </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #276018 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=276018\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E4-625%29%2F%28%28x%5E2-25%29%28x%2B5%29%28x%5E2%2B25%29%29+=+%28%28x%5E2-25%29%28x%5E2+%2B+25%29%29%2F%28%28x%5E2-25%29%28x%2B5%29%28x%5E2%2B25%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a)  If there are any cancellations, then the roots of the cancelled terms are candidates for the occurence of holes.  The domain of f is all real numbers except -5 and 5. So it's POSSIBLE for the function to have holes there.  \r<BR>\n" );
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document.write( "b)  The function becomes, after cancellation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E4-625%29%2F%28%28x%5E2-25%29%28x%2B5%29%28x%5E2%2B25%29%29+=+%28%28x%5E2-25%29%28x%5E2+%2B+25%29%29%2F%28%28x%5E2-25%29%28x%2B5%29%28x%5E2%2B25%29%29+=+1%2F%28x%2B5%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.  So there is a vertical asymptote at x = -5.\r<BR>\n" );
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document.write( "c)  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+25 ALIGN=MIDDLE  ALT=\"x%5E2+-+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > was cancelled from the top and the bottom.  Its roots are<BR>\n" );
document.write( "-5 and 5.  Since there is a vertical asymptote at x = -5, there is a \"hole\" at the graph only at x = 5.  The expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+25 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+25\"   DISABLED_style= \"max-width:90%; height:auto;\" >is never zero for real values of x.  (The roots are imaginary.)<BR>\n" );
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