document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=140329>Question 140329</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all the zeros of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E5-5x%5E4%2B3x%5E3-3x%5E2%2Bx%2B2 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E5-5x%5E4%2B3x%5E3-3x%5E2%2Bx%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<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 #102193 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=102193\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First graph the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E5-5x%5E4%2B3x%5E3-3x%5E2%2Bx%2B2 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E5-5x%5E4%2B3x%5E3-3x%5E2%2Bx%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From the graph, we can see that there is a zero 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;\" >. 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 function to the right of the test zero.<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><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></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 2)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><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 >2</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 2 by 2 and place the product (which is 4)  right underneath the second  coefficient (which is -5)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</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><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 4 and -5 to get -1. Place the sum right underneath 4.\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 2 by -1 and place the product (which is -2)  right underneath the third  coefficient (which is 3)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</TD><TD >-2</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -2 and 3 to get 1. Place the sum right underneath -2.\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</TD><TD >-2</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 2 by 1 and place the product (which is 2)  right underneath the fourth  coefficient (which is -3)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</TD><TD >-2</TD><TD >2</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 2 and -3 to get -1. Place the sum right underneath 2.\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</TD><TD >-2</TD><TD >2</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 2 by -1 and place the product (which is -2)  right underneath the fifth  coefficient (which is 1)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</TD><TD >-2</TD><TD >2</TD><TD >-2</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -2 and 1 to get -1. Place the sum right underneath -2.\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</TD><TD >-2</TD><TD >2</TD><TD >-2</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 2 by -1 and place the product (which is -2)  right underneath the sixth  coefficient (which is 2)\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</TD><TD >-2</TD><TD >2</TD><TD >-2</TD><TD >-2</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD><TD ></TD></TR>\n" );
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document.write( "    Add -2 and 2 to get 0. Place the sum right underneath -2.\r<BR>\n" );
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document.write( "<TR><TD >2</TD><TD >|</TD><TD >2</TD><TD >-5</TD><TD >3</TD><TD >-3</TD><TD >1</TD><TD >2</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >4</TD><TD >-2</TD><TD >2</TD><TD >-2</TD><TD >-2</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</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=2x%5E5+-+5x%5E4+%2B+3x%5E3+-+3x%5E2+%2B+x+%2B+2 ALIGN=MIDDLE  ALT=\"2x%5E5+-+5x%5E4+%2B+3x%5E3+-+3x%5E2+%2B+x+%2B+2\"   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 5 coefficients (2,-1,1,-1,-1) 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=%282x%5E5+-+5x%5E4+%2B+3x%5E3+-+3x%5E2+%2B+x+%2B+2%29%2F%28x-2%29=2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1 ALIGN=MIDDLE   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=2x%5E5+-+5x%5E4+%2B+3x%5E3+-+3x%5E2+%2B+x+%2B+2 ALIGN=MIDDLE  ALT=\"2x%5E5+-+5x%5E4+%2B+3x%5E3+-+3x%5E2+%2B+x+%2B+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%282x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1%29 ALIGN=MIDDLE  ALT=\"%28x-2%29%282x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1%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=2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1 ALIGN=MIDDLE  ALT=\"2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > down further\r<BR>\n" );
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document.write( "Now let's graph the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1\"   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=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1+++%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1+++%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From the graph, we can see that there is a zero at <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;\" >. So our test zero is 1. So this time 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 function to the right of the test zero.<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</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 2)\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</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 >2</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by 2 and place the product (which is 2)  right underneath the second  coefficient (which is -1)\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >2</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 2 and -1 to get 1. Place the sum right underneath 2.\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >2</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</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 third  coefficient (which is 1)\r<BR>\n" );
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document.write( "<TR><TD >1</TD><TD >|</TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >2</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 1 and 1 to get 2. 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 >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >2</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD >2</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply 1 by 2 and place the product (which is 2)  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 >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >2</TD><TD >1</TD><TD >2</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD >2</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 2 and -1 to get 1. Place the sum right underneath 2.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >1</TD><TD >|</TD><TD >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >2</TD><TD >1</TD><TD >2</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</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 fifth  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 >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\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 >2</TD><TD >-1</TD><TD >1</TD><TD >-1</TD><TD >-1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</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-1 ALIGN=MIDDLE  ALT=\"x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1 ALIGN=MIDDLE  ALT=\"2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1\"   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 (2,1,2,1) form the quotient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+%2B+x%5E2+%2B+2x+%2B+1 ALIGN=MIDDLE  ALT=\"2x%5E3+%2B+x%5E2+%2B+2x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1%29%2F%28x-1%29=2x%5E3+%2B+x%5E2+%2B+2x+%2B+1 ALIGN=MIDDLE  ALT=\"%282x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1%29%2F%28x-1%29=2x%5E3+%2B+x%5E2+%2B+2x+%2B+1\"   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=2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1 ALIGN=MIDDLE  ALT=\"2x%5E4+-+x%5E3+%2B+x%5E2+-+x+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%282x%5E3+%2B+x%5E2+%2B+2x+%2B+1%29 ALIGN=MIDDLE  ALT=\"%28x-1%29%282x%5E3+%2B+x%5E2+%2B+2x+%2B+1%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=2x%5E3+%2B+x%5E2+%2B+2x+%2B+1 ALIGN=MIDDLE  ALT=\"2x%5E3+%2B+x%5E2+%2B+2x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > down further\r<BR>\n" );
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document.write( "Now let's graph the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E3+%2B+x%5E2+%2B+2x+%2B+1 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E3+%2B+x%5E2+%2B+2x+%2B+1\"   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=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C2x%5E3+%2B+x%5E2+%2B+2x+%2B+1+++%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C2x%5E3+%2B+x%5E2+%2B+2x+%2B+1+++%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From the graph, we can see that there is a zero at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So our test zero is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this time 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 function to the right of the test zero.<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-1/2</TD><TD >|</TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</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 2)\r<BR>\n" );
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document.write( "<TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-1/2</TD><TD >|</TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</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 -1/2 by 2 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 >-1/2</TD><TD >|</TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</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 >2</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
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 >-1/2</TD><TD >|</TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</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 >2</TD><TD >0</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Multiply -1/2 by 0 and place the product (which is 0)  right underneath the third  coefficient (which is 2)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-1/2</TD><TD >|</TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</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 >2</TD><TD >0</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Add 0 and 2 to get 2. Place the sum right underneath 0.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-1/2</TD><TD >|</TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</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 >2</TD><TD >0</TD><TD >2</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
document.write( "    Multiply -1/2 by 2 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 >-1/2</TD><TD >|</TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-1</TD><TD >0</TD><TD >-1</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >0</TD><TD >2</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "\n" );
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 >-1/2</TD><TD >|</TD><TD >2</TD><TD >1</TD><TD >2</TD><TD >1</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-1</TD><TD >0</TD><TD >-1</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >0</TD><TD >2</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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B1 ALIGN=MIDDLE  ALT=\"2x%2B1\"   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+%2B+x%5E2+%2B+2x+%2B+1 ALIGN=MIDDLE  ALT=\"2x%5E3+%2B+x%5E2+%2B+2x+%2B+1\"   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 (2,0,2) form the quotient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+%2B+2 ALIGN=MIDDLE  ALT=\"2x%5E2+%2B+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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=2x%2B1 ALIGN=MIDDLE  ALT=\"2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the x term has a coefficient of 2, so we need to divide the quotient by 2 like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E2+%2B+2%29%2F2=x%5E2+%2B+1 ALIGN=MIDDLE  ALT=\"%282x%5E2+%2B+2%29%2F2=x%5E2+%2B+1\"   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=%282x%5E3+%2B+x%5E2+%2B+2x+%2B+1%29%2F%282x%2B1%29=x%5E2+%2B+1 ALIGN=MIDDLE  ALT=\"%282x%5E3+%2B+x%5E2+%2B+2x+%2B+1%29%2F%282x%2B1%29=x%5E2+%2B+1\"   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=2x%5E3+%2B+x%5E2+%2B+2x+%2B+1 ALIGN=MIDDLE  ALT=\"2x%5E3+%2B+x%5E2+%2B+2x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%2B1%29%28x%5E2+%2B+1%29 ALIGN=MIDDLE  ALT=\"%282x%2B1%29%28x%5E2+%2B+1%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=x%5E2+%2B+1 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > down further\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+1=0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the factor 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%5E2=-1 ALIGN=MIDDLE  ALT=\"x%5E2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 1 from both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0%2B-sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"x=0%2B-sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of both sides\r<BR>\n" );
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document.write( "Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=i ALIGN=MIDDLE  ALT=\"x=i\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-i ALIGN=MIDDLE  ALT=\"x=-i\"   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( "Answer:\r<BR>\n" );
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document.write( "So the zeros of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=2x%5E5-5x%5E4%2B3x%5E3-3x%5E2%2Bx%2B2 ALIGN=MIDDLE  ALT=\"f%28x%29=2x%5E5-5x%5E4%2B3x%5E3-3x%5E2%2Bx%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > are\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1%2F2 ALIGN=MIDDLE  ALT=\"x=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >, <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=i ALIGN=MIDDLE  ALT=\"x=i\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-i ALIGN=MIDDLE  ALT=\"x=-i\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );