document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=113804>Question 113804</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all the zeros of f (x) = 12x4 — 67x3 + 108x2 — 47x + 6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82821 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=82821\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> From part C), we found x=2 to be a zero for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E4-67x%5E3%2B108x%5E2-47x%2B6 ALIGN=MIDDLE  ALT=\"12x%5E4-67x%5E3%2B108x%5E2-47x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie f(2)=0) and the quotient was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E3+-+43x%5E2+%2B+22x+-+3 ALIGN=MIDDLE  ALT=\"12x%5E3+-+43x%5E2+%2B+22x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So let's find the zeros for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E3+-+43x%5E2+%2B+22x+-+3 ALIGN=MIDDLE  ALT=\"12x%5E3+-+43x%5E2+%2B+22x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E3+-+43x%5E2+%2B+22x+-+3 ALIGN=MIDDLE  ALT=\"12x%5E3+-+43x%5E2+%2B+22x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the same sign rules apply as they do to the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E4-67x%5E3%2B108x%5E2-47x%2B6 ALIGN=MIDDLE  ALT=\"12x%5E4-67x%5E3%2B108x%5E2-47x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >: there are no negative zeros and there are 4, 2, or no positive zeros\r<BR>\n" );
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document.write( "So let's find the possible zeros for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E3+-+43x%5E2+%2B+22x+-+3 ALIGN=MIDDLE  ALT=\"12x%5E3+-+43x%5E2+%2B+22x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Any rational zero can be found through this equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Roots=\frac{p}{q}\"> where p and q are the factors of the last and first coefficients\r<BR>\n" );
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document.write( "So let's list the factors of -3 (the last coefficient):\r<BR>\n" );
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document.write( "Now let's list the factors of 12 (the first coefficient):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q=\pm1, \pm2, \pm3, \pm4, \pm6, \pm12\">\r<BR>\n" );
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document.write( "Now let's divide each factor of the last coefficient by each factor of the first coefficient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{6}, \frac{1}{12}, \frac{3}{1}, \frac{3}{2}, \frac{3}{3}, \frac{3}{4}, \frac{3}{6}, \frac{3}{12}, \frac{-1}{1}, \frac{-1}{2}, \frac{-1}{3}, \frac{-1}{4}, \frac{-1}{6}, \frac{-1}{12}, \frac{-3}{1}, \frac{-3}{2}, \frac{-3}{3}, \frac{-3}{4}, \frac{-3}{6}, \frac{-3}{12}\">\r<BR>\n" );
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document.write( "Now simplify\r<BR>\n" );
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document.write( "These are all the distinct rational zeros of the function that could occur\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE  1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{6}, \frac{1}{12}, 3, \frac{3}{2}, \frac{3}{4}, -1, \frac{-1}{2}, \frac{-1}{3}, \frac{-1}{4}, \frac{-1}{6}, \frac{-1}{12}, -3, \frac{-3}{2}, \frac{-3}{4}\">\r<BR>\n" );
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document.write( "To save time, I'm only going to use synthetic division on the possible positive  zeros (using Descartes rule of signs) that are actually zeros of the function.<BR>\n" );
document.write( "Otherwise, I would have to use synthetic division on every possible positive root (there are 12 possible positive roots, so that means there would be at most 12 synthetic division tables).<BR>\n" );
document.write( "However, you might be required to follow this procedure, so this is why I'm showing you how to set up a problem like this\r<BR>\n" );
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document.write( "If you're not required to follow this procedure, simply use a graphing calculator to find the roots\r<BR>\n" );
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document.write( "So let's use the zero x=3 (which is an actual zero)\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 >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><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 12)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 3 by 12 and place the product (which is 36)  right underneath the second  coefficient (which is -43)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >36</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 36 and -43 to get -7. Place the sum right underneath 36.\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >36</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-7</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 3 by -7 and place the product (which is -21)  right underneath the third  coefficient (which is 22)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >36</TD><TD >-21</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-7</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -21 and 22 to get 1. Place the sum right underneath -21.\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >36</TD><TD >-21</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-7</TD><TD >1</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 fourth  coefficient (which is -3)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >36</TD><TD >-21</TD><TD >3</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-7</TD><TD >1</TD><TD ></TD></TR>\n" );
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document.write( "    Add 3 and -3 to get 0. Place the sum right underneath 3.\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >36</TD><TD >-21</TD><TD >3</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-7</TD><TD >1</TD><TD >0</TD></TR>\n" );
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document.write( "Since the last column adds to zero, we have a remainder of zero. 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 a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E3+-+43x%5E2+%2B+22x+-+3 ALIGN=MIDDLE  ALT=\"12x%5E3+-+43x%5E2+%2B+22x+-+3\"   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 3 coefficients (12,-7,1) form the quotient\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812x%5E3+-+43x%5E2+%2B+22x+-+3%29%2F%28x-3%29=12x%5E2+-+7x+%2B+1 ALIGN=MIDDLE  ALT=\"%2812x%5E3+-+43x%5E2+%2B+22x+-+3%29%2F%28x-3%29=12x%5E2+-+7x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can use this <a  rel=nofollow HREF=http://calc101.com/webMathematica/long-divide.jsp>online polynomial division calculator</a> to check your work\r<BR>\n" );
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document.write( "Basically  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E3+-+43x%5E2+%2B+22x+-+3 ALIGN=MIDDLE  ALT=\"12x%5E3+-+43x%5E2+%2B+22x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%2812x%5E2+-+7x+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x-3%29%2812x%5E2+-+7x+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets break  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2+-+7x+%2B+1 ALIGN=MIDDLE  ALT=\"12x%5E2+-+7x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > down further\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E2+-+7x+%2B+1=0 ALIGN=MIDDLE  ALT=\"12x%5E2+-+7x+%2B+1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the expression equal to zero and solve for x\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for x:\r<BR>\n" );
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document.write( "Starting with the general quadratic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the general solution using the quadratic equation is:\r<BR>\n" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2Ax%5E2-7%2Ax%2B1=0 ALIGN=MIDDLE  ALT=\"12%2Ax%5E2-7%2Ax%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=12 ALIGN=MIDDLE  ALT=\"a=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-7 ALIGN=MIDDLE  ALT=\"b=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=1 ALIGN=MIDDLE  ALT=\"c=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28--7+%2B-+sqrt%28+%28-7%29%5E2-4%2A12%2A1+%29%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%28--7+%2B-+sqrt%28+%28-7%29%5E2-4%2A12%2A1+%29%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=12, b=-7, and c=1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%287+%2B-+sqrt%28+%28-7%29%5E2-4%2A12%2A1+%29%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%287+%2B-+sqrt%28+%28-7%29%5E2-4%2A12%2A1+%29%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate -7 to get 7\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%287+%2B-+sqrt%28+49-4%2A12%2A1+%29%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%287+%2B-+sqrt%28+49-4%2A12%2A1+%29%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -7 to get 49  (note: remember when you square -7, you must square the negative as well. This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-7%29%5E2=-7%2A-7=49 ALIGN=MIDDLE  ALT=\"%28-7%29%5E2=-7%2A-7=49\"   DISABLED_style= \"max-width:90%; height:auto;\" >.)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%287+%2B-+sqrt%28+1+%29%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%287+%2B-+sqrt%28+1+%29%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%287+%2B-+1%29%2F%282%2A12%29 ALIGN=MIDDLE  ALT=\"x+=+%287+%2B-+1%29%2F%282%2A12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%287+%2B-+1%29%2F24 ALIGN=MIDDLE  ALT=\"x+=+%287+%2B-+1%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 12 to get 24\r<BR>\n" );
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document.write( "So now the expression breaks down into two parts\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%287+%2B+1%29%2F24 ALIGN=MIDDLE  ALT=\"x+=+%287+%2B+1%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%287+-+1%29%2F24 ALIGN=MIDDLE  ALT=\"x+=+%287+-+1%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Lets look at the first part:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%287+%2B+1%29%2F24 ALIGN=MIDDLE  ALT=\"x=%287+%2B+1%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=8%2F24 ALIGN=MIDDLE  ALT=\"x=8%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add the terms in the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F3 ALIGN=MIDDLE  ALT=\"x=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So one answer is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F3 ALIGN=MIDDLE  ALT=\"x=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets look at the second part:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%287+-+1%29%2F24 ALIGN=MIDDLE  ALT=\"x=%287+-+1%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6%2F24 ALIGN=MIDDLE  ALT=\"x=6%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F4 ALIGN=MIDDLE  ALT=\"x=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide\r<BR>\n" );
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document.write( "So another answer is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F4 ALIGN=MIDDLE  ALT=\"x=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the solutions for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2Ax%5E2-7%2Ax%2B1 ALIGN=MIDDLE  ALT=\"12%2Ax%5E2-7%2Ax%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F3 ALIGN=MIDDLE  ALT=\"x=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F4 ALIGN=MIDDLE  ALT=\"x=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "========================================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the zeros are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F3 ALIGN=MIDDLE  ALT=\"x=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1%2F4 ALIGN=MIDDLE  ALT=\"x=1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >,  <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( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );