document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=113803>Question 113803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Divide f (x) = 12x4 — 67x3 + 108x2 — 47x + 6 by p(x) = x — 2, to f ind the quotient.<BR>\n" );
document.write( "Use the remainder theorem to confirm you results<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 #82819 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=82819\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's simplify this expression 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=%2812x%5E4+-+67x%5E3+%2B+108x%5E2+-+47x+%2B+6%29%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"%2812x%5E4+-+67x%5E3+%2B+108x%5E2+-+47x+%2B+6%29%2F%28x-2%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-2=0 ALIGN=MIDDLE  ALT=\"x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator <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;\" > 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=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for x.\r<BR>\n" );
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document.write( "so our test zero is 2\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 >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</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 12)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</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 >12</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 2 by 12 and place the product (which is 24)  right underneath the second  coefficient (which is -67)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >24</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 24 and -67 to get -43. Place the sum right underneath 24.\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >24</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-43</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 2 by -43 and place the product (which is -86)  right underneath the third  coefficient (which is 108)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >24</TD><TD >-86</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-43</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -86 and 108 to get 22. Place the sum right underneath -86.\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >24</TD><TD >-86</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 2 by 22 and place the product (which is 44)  right underneath the fourth  coefficient (which is -47)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >24</TD><TD >-86</TD><TD >44</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 44 and -47 to get -3. Place the sum right underneath 44.\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >24</TD><TD >-86</TD><TD >44</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 2 by -3 and place the product (which is -6)  right underneath the fifth  coefficient (which is 6)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >24</TD><TD >-86</TD><TD >44</TD><TD >-6</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</TD><TD ></TD></TR>\n" );
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document.write( "    Add -6 and 6 to get 0. Place the sum right underneath -6.\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >12</TD><TD >-67</TD><TD >108</TD><TD >-47</TD><TD >6</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >24</TD><TD >-86</TD><TD >44</TD><TD >-6</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >12</TD><TD >-43</TD><TD >22</TD><TD >-3</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-2 ALIGN=MIDDLE  ALT=\"x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x%5E4+-+67x%5E3+%2B+108x%5E2+-+47x+%2B+6 ALIGN=MIDDLE  ALT=\"12x%5E4+-+67x%5E3+%2B+108x%5E2+-+47x+%2B+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 4 coefficients (12,-43,22,-3) 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%5E4+-+67x%5E3+%2B+108x%5E2+-+47x+%2B+6%29%2F%28x-2%29=12x%5E3+-+43x%5E2+%2B+22x+-+3 ALIGN=MIDDLE  ALT=\"%2812x%5E4+-+67x%5E3+%2B+108x%5E2+-+47x+%2B+6%29%2F%28x-2%29=12x%5E3+-+43x%5E2+%2B+22x+-+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( "Summary:\r<BR>\n" );
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document.write( "So our quotient is <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( "Our remainder is zero\r<BR>\n" );
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document.write( "Since our remainder is zero, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29=0 ALIGN=MIDDLE  ALT=\"f%282%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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