document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=89876>Question 89876</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What are all the rational zeros of these polynomials?y=2x^3-10x^2-8x+40 and y=3x^3-24x^2-48x+384 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #65249 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=65249\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \"What are the rational zeros of y=2x^3-10x^2-8x+40?\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Any rational zero can be found through this equation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Roots=\frac{p}{q}\"> where p and q are the combinations of the last and first coefficients\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So let's list the factors of p (which is 40)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p=\pm1, \pm2, \pm4, \pm5, \pm8, \pm10, \pm20, \pm40\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's list the factors of q (which is 2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q=\pm1, \pm2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's divide them\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{2}{2}, \frac{4}{1}, \frac{4}{2}, \frac{5}{1}, \frac{5}{2}, \frac{8}{1}, \frac{8}{2}, \frac{10}{1}, \frac{10}{2}, \frac{20}{1}, \frac{20}{2}, \frac{40}{1}, \frac{40}{2}, \frac{-1}{1}, \frac{-1}{2}, \frac{-2}{1}, \frac{-2}{2}, \frac{-4}{1}, \frac{-4}{2}, \frac{-5}{1}, \frac{-5}{2}, \frac{-8}{1}, \frac{-8}{2}, \frac{-10}{1}, \frac{-10}{2}, \frac{-20}{1}, \frac{-20}{2}, \frac{-40}{1}, \frac{-40}{2}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "These are all the possible zeros of the function\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1, \frac{1}{2}, 2, 1, 4, 2, 5, \frac{5}{2}, 8, 4, 10, 5, 20, 10, 40, 20, -1, \frac{-1}{2}, -2, -1, -4, -2, -5, \frac{-5}{2}, -8, -4, -10, -5, -20, -10, -40, -20\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To save time, I'm only going to use synthetic division on the possible zeros that are actually zeros of the function.<BR>\n" );
document.write( "Otherwise, I would have to use synthetic division on every possible root (there are 32 possible roots, so that means there would be at most 32 synthetic division tables).<BR>\n" );
document.write( "However, you might be required to follow this procedure, so this is why I'm using this procedure\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you're not required to follow this procedure, simply use a graphing calculator to find the roots\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we find that the equation has a root at <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;\" >. This means the test zero is -2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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 >-2</TD><TD >|</TD><TD >2</TD><TD >-10</TD><TD >-8</TD><TD >40</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" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 2)\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-2</TD><TD >|</TD><TD >2</TD><TD >-10</TD><TD >-8</TD><TD >40</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Multiply -2 by 2 and place the product (which is -4)  right underneath the second  coefficient (which is -10)\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-2</TD><TD >|</TD><TD >2</TD><TD >-10</TD><TD >-8</TD><TD >40</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Add -4 and -10 to get -14. Place the sum right underneath -4.\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-2</TD><TD >|</TD><TD >2</TD><TD >-10</TD><TD >-8</TD><TD >40</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-14</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Multiply -2 by -14 and place the product (which is 28)  right underneath the third  coefficient (which is -8)\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-2</TD><TD >|</TD><TD >2</TD><TD >-10</TD><TD >-8</TD><TD >40</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD >28</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-14</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Add 28 and -8 to get 20. Place the sum right underneath 28.\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-2</TD><TD >|</TD><TD >2</TD><TD >-10</TD><TD >-8</TD><TD >40</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD >28</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-14</TD><TD >20</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Multiply -2 by 20 and place the product (which is -40)  right underneath the fourth  coefficient (which is 40)\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-2</TD><TD >|</TD><TD >2</TD><TD >-10</TD><TD >-8</TD><TD >40</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD >28</TD><TD >-40</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-14</TD><TD >20</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Add -40 and 40 to get 0. Place the sum right underneath -40.\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-2</TD><TD >|</TD><TD >2</TD><TD >-10</TD><TD >-8</TD><TD >40</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD >28</TD><TD >-40</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-14</TD><TD >20</TD><TD >0</TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
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%2B2 ALIGN=MIDDLE  ALT=\"x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+-+10x%5E2+-+8x+%2B+40 ALIGN=MIDDLE  ALT=\"2x%5E3+-+10x%5E2+-+8x+%2B+40\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now lets look at the bottom row of coefficients:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first 3 coefficients (2,-14,20) form the quotient\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+14x+%2B+20 ALIGN=MIDDLE  ALT=\"2x%5E2+-+14x+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E3+-+10x%5E2+-+8x+%2B+40%29%2F%28x%2B2%29=2x%5E2+-+14x+%2B+20 ALIGN=MIDDLE  ALT=\"%282x%5E3+-+10x%5E2+-+8x+%2B+40%29%2F%28x%2B2%29=2x%5E2+-+14x+%2B+20\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So that means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3-10x%5E2-8x%2B40 ALIGN=MIDDLE  ALT=\"2x%5E3-10x%5E2-8x%2B40\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%282x%5E2+-+14x+%2B+20%29 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%282x%5E2+-+14x+%2B+20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%28x-5%29%282x-4%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%28x-5%29%282x-4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor completely\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the zeros are \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<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=5 ALIGN=MIDDLE  ALT=\"x=5\"   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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "==========================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "\"What are the rational zeros of y=3x^3-24x^2-48x+384?\"\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Any rational zero can be found through this equation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE Roots=\frac{p}{q}\"> where p and q are the combinations of the last and first coefficients\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So let's list the factors of p (which is 384)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p=\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm12, \pm16, \pm24, \pm32, \pm48, \pm64, \pm96, \pm128, \pm192, \pm384\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's list the factors of q (which is 3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q=\pm1, \pm3\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's divide them\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{1}, \frac{1}{3}, \frac{2}{1}, \frac{2}{3}, \frac{3}{1}, \frac{3}{3}, \frac{4}{1}, \frac{4}{3}, \frac{6}{1}, \frac{6}{3}, \frac{8}{1}, \frac{8}{3}, \frac{12}{1}, \frac{12}{3}, \frac{16}{1}, \frac{16}{3}, \frac{24}{1}, \frac{24}{3}, \frac{32}{1}, \frac{32}{3}, \frac{48}{1}, \frac{48}{3}, \frac{64}{1}, \frac{64}{3}, \frac{96}{1}, \frac{96}{3}, \frac{128}{1}, \frac{128}{3}, \frac{192}{1}, \frac{192}{3}, \frac{384}{1}, \frac{384}{3}, \frac{-1}{1}, \frac{-1}{3}, \frac{-2}{1}, \frac{-2}{3}, \frac{-3}{1}, \frac{-3}{3}, \frac{-4}{1}, \frac{-4}{3}, \frac{-6}{1}, \frac{-6}{3}, \frac{-8}{1}, \frac{-8}{3}, \frac{-12}{1}, \frac{-12}{3}, \frac{-16}{1}, \frac{-16}{3}, \frac{-24}{1}, \frac{-24}{3}, \frac{-32}{1}, \frac{-32}{3}, \frac{-48}{1}, \frac{-48}{3}, \frac{-64}{1}, \frac{-64}{3}, \frac{-96}{1}, \frac{-96}{3}, \frac{-128}{1}, \frac{-128}{3}, \frac{-192}{1}, \frac{-192}{3}, \frac{-384}{1}, \frac{-384}{3}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "These are all the possible zeros of the function\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 1, \frac{1}{3}, 2, \frac{2}{3}, 3, 1, 4, \frac{4}{3}, 6, 2, 8, \frac{8}{3}, 12, 4, 16, \frac{16}{3}, 24, 8, 32, \frac{32}{3}, 48, 16, 64, \frac{64}{3}, 96, 32, 128, \frac{128}{3}, 192, 64, 384, 128, -1, \frac{-1}{3}, -2, \frac{-2}{3}, -3, -1, -4, \frac{-4}{3}, -6, -2, -8, \frac{-8}{3}, -12, -4, -16, \frac{-16}{3}, -24, -8, -32, \frac{-32}{3}, -48, -16, -64, \frac{-64}{3}, -96, -32, -128, \frac{-128}{3}, -192, -64, -384, -128\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To save time, I'm only going to use synthetic division on the possible zeros that are actually zeros of the function.<BR>\n" );
document.write( "Otherwise, I would have to use synthetic division on every possible root (there are 64 possible roots, so that means there would be at most 64 synthetic division tables).<BR>\n" );
document.write( "However, you might be required to follow this procedure, so this is why I'm using this procedure\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you're not required to follow this procedure, simply use a graphing calculator to find the roots\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we find that the equation has a root at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This means the test zero is -4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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 >-4</TD><TD >|</TD><TD >3</TD><TD >-24</TD><TD >-48</TD><TD >384</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" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 3)\r<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >3</TD><TD >-24</TD><TD >-48</TD><TD >384</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Multiply -4 by 3 and place the product (which is -12)  right underneath the second  coefficient (which is -24)\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >3</TD><TD >-24</TD><TD >-48</TD><TD >384</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-12</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Add -12 and -24 to get -36. Place the sum right underneath -12.\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >3</TD><TD >-24</TD><TD >-48</TD><TD >384</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-12</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >-36</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Multiply -4 by -36 and place the product (which is 144)  right underneath the third  coefficient (which is -48)\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >3</TD><TD >-24</TD><TD >-48</TD><TD >384</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-12</TD><TD >144</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >-36</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Add 144 and -48 to get 96. Place the sum right underneath 144.\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >3</TD><TD >-24</TD><TD >-48</TD><TD >384</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-12</TD><TD >144</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >-36</TD><TD >96</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Multiply -4 by 96 and place the product (which is -384)  right underneath the fourth  coefficient (which is 384)\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >3</TD><TD >-24</TD><TD >-48</TD><TD >384</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-12</TD><TD >144</TD><TD >-384</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >-36</TD><TD >96</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Add -384 and 384 to get 0. Place the sum right underneath -384.\r<BR>\n" );
document.write( "\n" );
document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >3</TD><TD >-24</TD><TD >-48</TD><TD >384</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-12</TD><TD >144</TD><TD >-384</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >-36</TD><TD >96</TD><TD >0</TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
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%2B4 ALIGN=MIDDLE  ALT=\"x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E3+-+24x%5E2+-+48x+%2B+384 ALIGN=MIDDLE  ALT=\"3x%5E3+-+24x%5E2+-+48x+%2B+384\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now lets look at the bottom row of coefficients:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first 3 coefficients (3,-36,96) form the quotient\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+-+36x+%2B+96 ALIGN=MIDDLE  ALT=\"3x%5E2+-+36x+%2B+96\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E3+-+24x%5E2+-+48x+%2B+384%29%2F%28x%2B4%29=3x%5E2+-+36x+%2B+96 ALIGN=MIDDLE  ALT=\"%283x%5E3+-+24x%5E2+-+48x+%2B+384%29%2F%28x%2B4%29=3x%5E2+-+36x+%2B+96\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So that means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E3+-+24x%5E2+-+48x+%2B+384 ALIGN=MIDDLE  ALT=\"3x%5E3+-+24x%5E2+-+48x+%2B+384\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%283x%5E2+-+36x+%2B+96%29 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%283x%5E2+-+36x+%2B+96%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%28x-4%29%283x-24%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%28x-4%29%283x-24%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor completely\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the zeros are \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-4 ALIGN=MIDDLE  ALT=\"x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=8 ALIGN=MIDDLE  ALT=\"x=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );