document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=135713>Question 135713</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use synthetic division grid to analyze and graph the function.<BR>\n" );
document.write( "     P(x) = 9x^4 – 6x^3 – 26x^2 + 18 x - 3  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #99460 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=99460\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 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 9 (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, \pm3, \pm9\">\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}{3}, \frac{1}{9}, \frac{3}{1}, \frac{3}{3}, \frac{3}{9}, \frac{-1}{1}, \frac{-1}{3}, \frac{-1}{9}, \frac{-3}{1}, \frac{-3}{3}, \frac{-3}{9}\">\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}{3}, \frac{1}{9}, 3, -1, \frac{-1}{3}, \frac{-1}{9}, -3\">\r<BR>\n" );
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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 8 possible roots, so that means there would be at most 8 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( "It turns out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is actually a root. So our test zero is 1/3\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 >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><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><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 9)\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >9</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1/3 by 9 and place the product (which is 3)  right underneath the second  coefficient (which is -6)\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >9</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 3 and -6 to get -3. Place the sum right underneath 3.\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >9</TD><TD >-3</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1/3 by -3 and place the product (which is -1)  right underneath the third  coefficient (which is -26)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >-1</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >9</TD><TD >-3</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -1 and -26 to get -27. Place the sum right underneath -1.\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >-1</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >9</TD><TD >-3</TD><TD >-27</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1/3 by -27 and place the product (which is -9)  right underneath the fourth  coefficient (which is 18)\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >9</TD><TD >-3</TD><TD >-27</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -9 and 18 to get 9. Place the sum right underneath -9.\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >9</TD><TD >-3</TD><TD >-27</TD><TD >9</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1/3 by 9 and place the product (which is 3)  right underneath the fifth  coefficient (which is -3)\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD >3</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >9</TD><TD >-3</TD><TD >-27</TD><TD >9</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 >1/3</TD><TD >|</TD><TD >9</TD><TD >-6</TD><TD >-26</TD><TD >18</TD><TD >-3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD >3</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >9</TD><TD >-3</TD><TD >-27</TD><TD >9</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=3x-1 ALIGN=MIDDLE  ALT=\"3x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+3 ALIGN=MIDDLE  ALT=\"9x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+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 4 coefficients (9,-3,-27,9) form the quotient\r<BR>\n" );
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document.write( "Notice in the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-1 ALIGN=MIDDLE  ALT=\"3x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the x term has a coefficient of 3, so we need to divide the quotient by 3 like this:\r<BR>\n" );
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document.write( "  \r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+3%29%2F%283x-1%29=3x%5E3+-+x%5E2+-+9x+%2B+3 ALIGN=MIDDLE  ALT=\"%289x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+3%29%2F%283x-1%29=3x%5E3+-+x%5E2+-+9x+%2B+3\"   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=9x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+3 ALIGN=MIDDLE  ALT=\"9x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x-1%29%283x%5E3+-+x%5E2+-+9x+%2B+3%29 ALIGN=MIDDLE  ALT=\"%283x-1%29%283x%5E3+-+x%5E2+-+9x+%2B+3%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=3x%5E3+-+x%5E2+-+9x+%2B+3 ALIGN=MIDDLE  ALT=\"3x%5E3+-+x%5E2+-+9x+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > down further\r<BR>\n" );
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document.write( "Once again through the rational root theorem, we find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is another root (since it occurs twice)\r<BR>\n" );
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document.write( "So once again our test zero is 1/3\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 >1/3</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-9</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 3)\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-9</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 >3</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1/3 by 3 and place the product (which is 1)  right underneath the second  coefficient (which is -1)\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD >3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 1 and -1 to get 0. Place the sum right underneath 1.\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD >3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >0</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1/3 by 0 and place the product (which is 0)  right underneath the third  coefficient (which is -9)\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD >3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >0</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 0 and -9 to get -9. Place the sum right underneath 0.\r<BR>\n" );
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document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD >3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >0</TD><TD >-9</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1/3 by -9 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 >1/3</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD >3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >-3</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >0</TD><TD >-9</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( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1/3</TD><TD >|</TD><TD >3</TD><TD >-1</TD><TD >-9</TD><TD >3</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >-3</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >3</TD><TD >0</TD><TD >-9</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=3x-1 ALIGN=MIDDLE  ALT=\"3x-1\"   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+-+x%5E2+-+9x+%2B+3 ALIGN=MIDDLE  ALT=\"3x%5E3+-+x%5E2+-+9x+%2B+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 (3,0,-9) form the quotient\r<BR>\n" );
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document.write( "Notice in the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-1 ALIGN=MIDDLE  ALT=\"3x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the x term has a coefficient of 3, so we need to divide the quotient by 3 like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E2+-+9%29%2F2=x%5E2+-+3 ALIGN=MIDDLE  ALT=\"%283x%5E2+-+9%29%2F2=x%5E2+-+3\"   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( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%5E3+-+x%5E2+-+9x+%2B+3%29%2F%283x-1%29=x%5E2+-+3 ALIGN=MIDDLE  ALT=\"%283x%5E3+-+x%5E2+-+9x+%2B+3%29%2F%283x-1%29=x%5E2+-+3\"   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( "So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+3 ALIGN=MIDDLE  ALT=\"9x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-3%29%283x-1%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%5E2-3%29%283x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This tells us that the zeros are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%283%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%283%29 ALIGN=MIDDLE  ALT=\"x=sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and  <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;\" > (with a multiplicity of 2)\r<BR>\n" );
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document.write( "So we can now graph the function by drawing a curve through the zeros\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+9x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+3%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+9x%5E4+-+6x%5E3+-+26x%5E2+%2B+18x+-+3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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