document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=149767>Question 149767</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the remainder when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E7-x%5E6%2Bx%5E5-x%5E4%2B4 ALIGN=MIDDLE  ALT=\"x%5E7-x%5E6%2Bx%5E5-x%5E4%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divided by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1 ALIGN=MIDDLE  ALT=\"x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #109866 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=109866\">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=%28x%5E7+-+x%5E6+%2B+x%5E5+-+x%5E4+%2B+4%29%2F%28x-1%29 ALIGN=MIDDLE  ALT=\"%28x%5E7+-+x%5E6+%2B+x%5E5+-+x%5E4+%2B+4%29%2F%28x-1%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-1=0 ALIGN=MIDDLE  ALT=\"x-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1 ALIGN=MIDDLE  ALT=\"x-1\"   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=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for x.\r<BR>\n" );
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document.write( "so our test zero is 1\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.(note: remember if a polynomial goes from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E4 ALIGN=MIDDLE  ALT=\"-x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > there is a zero coefficient for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is simply because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E7+-+x%5E6+%2B+x%5E5+-+x%5E4+%2B+4 ALIGN=MIDDLE  ALT=\"x%5E7+-+x%5E6+%2B+x%5E5+-+x%5E4+%2B+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > really looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E7%2B-1x%5E6%2B1x%5E5%2B-1x%5E4%2B0x%5E3%2B0x%5E2%2B0x%5E1%2B4x%5E0 ALIGN=MIDDLE  ALT=\"1x%5E7%2B-1x%5E6%2B1x%5E5%2B-1x%5E4%2B0x%5E3%2B0x%5E2%2B0x%5E1%2B4x%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><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><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 1)\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by 1 and place the product (which is 1)  right underneath the second  coefficient (which is -1)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></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( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by 0 and place the product (which is 0)  right underneath the third  coefficient (which is 1)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Add 0 and 1 to get 1. Place the sum right underneath 0.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Multiply 1 by 1 and place the product (which is 1)  right underneath the fourth  coefficient (which is -1)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD ></TD><TD ></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( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Multiply 1 by 0 and place the product (which is 0)  right underneath the fifth  coefficient (which is 0)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 0 and 0 to get 0. Place the sum right underneath 0.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by 0 and place the product (which is 0)  right underneath the sixth  coefficient (which is 0)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 0 and 0 to get 0. Place the sum right underneath 0.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by 0 and place the product (which is 0)  right underneath the seventh  coefficient (which is 0)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 0 and 0 to get 0. Place the sum right underneath 0.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by 0 and place the product (which is 0)  right underneath the eighth  coefficient (which is 4)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD ></TD></TR>\n" );
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document.write( "    Add 0 and 4 to get 4. Place the sum right underneath 0.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >1</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >0</TD><TD >1</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >0</TD><TD >4</TD></TR>\n" );
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document.write( "Since the last column adds to 4, we have a remainder of 4. \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E7-x%5E6%2Bx%5E5-x%5E4%2B4 ALIGN=MIDDLE  ALT=\"x%5E7-x%5E6%2Bx%5E5-x%5E4%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divided by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-1 ALIGN=MIDDLE  ALT=\"x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the remainder is 4 </SPAN>\n" );
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