document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=147748>Question 147748</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use synthetic division to show that the number given to the right of the equation is a solution of the equation, then solve the polynomial equation.<BR>\n" );
document.write( "x^3+2x^2-11x-12=0;-4\r<BR>\n" );
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document.write( "a){3,1,-4}<BR>\n" );
document.write( "b){3.-1.-4}<BR>\n" );
document.write( "c){-3,1,-4)<BR>\n" );
document.write( "d){-3,-1,-4} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #108116 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=108116\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "First set up the synthetic division table by placing the test zero (which in this case is -4) in the upper left corner and placing the coefficients of the function to the right of the test zero.<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >-11</TD><TD >-12</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 1)\r<BR>\n" );
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document.write( "<TR><TD >-4</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >-11</TD><TD >-12</TD></TR><TR><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></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Multiply -4 by 1 and place the product (which is -4)  right underneath the second  coefficient (which is 2)\r<BR>\n" );
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document.write( "<TR><TD >-4</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >-11</TD><TD >-12</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 >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Add -4 and 2 to get -2. Place the sum right underneath -4.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >-11</TD><TD >-12</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 >1</TD><TD >-2</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Multiply -4 by -2 and place the product (which is 8)  right underneath the third  coefficient (which is -11)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >-11</TD><TD >-12</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD >8</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >-2</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Add 8 and -11 to get -3. Place the sum right underneath 8.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >-11</TD><TD >-12</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD >8</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >-2</TD><TD >-3</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Multiply -4 by -3 and place the product (which is 12)  right underneath the fourth  coefficient (which is -12)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >-11</TD><TD >-12</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD >8</TD><TD >12</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >-2</TD><TD >-3</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Add 12 and -12 to get 0. Place the sum right underneath 12.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-4</TD><TD >|</TD><TD >1</TD><TD >2</TD><TD >-11</TD><TD >-12</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-4</TD><TD >8</TD><TD >12</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >-2</TD><TD >-3</TD><TD >0</TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "Since the last column adds to zero, we have a remainder of zero. This means that -4 is a solution of the equation\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 (1,-2,-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=%28x%5E3+%2B+2x%5E2+-+11x+-+12%29%2F%28x%2B4%29=x%5E2+-+2x+-+3 ALIGN=MIDDLE  ALT=\"%28x%5E3+%2B+2x%5E2+-+11x+-+12%29%2F%28x%2B4%29=x%5E2+-+2x+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Basically  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+%2B+2x%5E2+-+11x+-+12 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+2x%5E2+-+11x+-+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%28x%5E2+-+2x+-+3%29 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%28x%5E2+-+2x+-+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets solve  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2x+-+3=0 ALIGN=MIDDLE  ALT=\"x%5E2+-+2x+-+3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the quadratic formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-3%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-3%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=-3 ALIGN=MIDDLE  ALT=\"c=-3\"   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+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-3%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%281%29%28-3%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   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+=+%282+%2B-+sqrt%28+4-4%281%29%28-3%29+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4-4%281%29%28-3%29+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   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+=+%282+%2B-+sqrt%28+4--12+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4--12+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281%29%28-3%29 ALIGN=MIDDLE  ALT=\"4%281%29%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   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+=+%282+%2B-+sqrt%28+4%2B12+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+4%2B12+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284--12%29 ALIGN=MIDDLE  ALT=\"sqrt%284--12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%2B12%29 ALIGN=MIDDLE  ALT=\"sqrt%284%2B12%29\"   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+=+%282+%2B-+sqrt%28+16+%29%29%2F%282%281%29%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+16+%29%29%2F%282%281%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   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+=+%282+%2B-+sqrt%28+16+%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+sqrt%28+16+%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   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+=+%282+%2B-+4%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B-+4%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   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+=+%282+%2B+4%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%282+%2B+4%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282+-+4%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%282+-+4%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the expression. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%286%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=+%286%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=++%28-2%29%2F%282%29 ALIGN=MIDDLE  ALT=\"x+=++%28-2%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms. \r<BR>\n" );
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document.write( "<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;\" > or <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;\" > Simplify. \r<BR>\n" );
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document.write( "So the answers are:\r<BR>\n" );
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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+=+3 ALIGN=MIDDLE  ALT=\"x+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <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;\" > \r<BR>\n" );
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document.write( " </SPAN>\n" );
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