document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=85753>Question 85753</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dividing Polynomials: synthetic\r<BR>\n" );
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document.write( "2x^4 - 10x^2 + 6x - 8 divided by x + 3\r<BR>\n" );
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document.write( "please help me if you can </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #61974 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=61974\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "Start with the given polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E4+-+10x%5E2+%2B+6x+-+8%29%2F%28x%2B3%29 ALIGN=MIDDLE  ALT=\"%282x%5E4+-+10x%5E2+%2B+6x+-+8%29%2F%28x%2B3%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%2B3=0 ALIGN=MIDDLE  ALT=\"x%2B3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3 ALIGN=MIDDLE  ALT=\"x%2B3\"   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=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for x.\r<BR>\n" );
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document.write( "so our test zero is -3\r<BR>\n" );
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document.write( "Now set up the synthetic division table by placing the test zero in the upperleft 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=2x%5E4 ALIGN=MIDDLE  ALT=\"2x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10x%5E2 ALIGN=MIDDLE  ALT=\"-10x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > there is a zero coefficient. This is simply because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E4+-+10x%5E2+%2B+6x+-+8 ALIGN=MIDDLE  ALT=\"2x%5E4+-+10x%5E2+%2B+6x+-+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > really looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E4%2B0x%5E3%2B-10x%5E2%2B6x%5E1%2B-8x%5E0 ALIGN=MIDDLE  ALT=\"2x%5E4%2B0x%5E3%2B-10x%5E2%2B6x%5E1%2B-8x%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</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 >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</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 -3 by 2 and place the product (which is -6)  right underneath 0\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD >-6</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 -6 and 0 to get -6. Place the sum right underneath -6.\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD >-6</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-6</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -3 by -6 and place the product (which is 18)  right underneath -10\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD >-6</TD><TD >18</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-6</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add 18 and -10 to get 8. Place the sum right underneath 18.\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD >-6</TD><TD >18</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-6</TD><TD >8</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -3 by 8 and place the product (which is -24)  right underneath 6\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD >-6</TD><TD >18</TD><TD >-24</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-6</TD><TD >8</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -24 and 6 to get -18. Place the sum right underneath -24.\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD >-6</TD><TD >18</TD><TD >-24</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-6</TD><TD >8</TD><TD >-18</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -3 by -18 and place the product (which is 54)  right underneath -8\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD >-6</TD><TD >18</TD><TD >-24</TD><TD >54</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-6</TD><TD >8</TD><TD >-18</TD><TD ></TD></TR>\n" );
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document.write( "    Add 54 and -8 to get 46. Place the sum right underneath 54.\r<BR>\n" );
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document.write( "<TR><TD >-3</TD><TD >|</TD><TD >2</TD><TD >0</TD><TD >-10</TD><TD >6</TD><TD >-8</TD></TR><TR><TD ></TD><TD ></TD><TD ></TD><TD >-6</TD><TD >18</TD><TD >-24</TD><TD >54</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >2</TD><TD >-6</TD><TD >8</TD><TD >-18</TD><TD >46</TD></TR>\n" );
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document.write( "Since the last column adds to 46, we have a remainder of 46. This means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3 ALIGN=MIDDLE  ALT=\"x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a <b>not</b> factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E4+-+10x%5E2+%2B+6x+-+8 ALIGN=MIDDLE  ALT=\"2x%5E4+-+10x%5E2+%2B+6x+-+8\"   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,-6,8,-18) 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+-+6x%5E2+%2B+8x+-+18 ALIGN=MIDDLE  ALT=\"2x%5E3+-+6x%5E2+%2B+8x+-+18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and the last coefficient 46, is the remainder, which is placed over <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3 ALIGN=MIDDLE  ALT=\"x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > like this\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=46%2F%28x%2B3%29 ALIGN=MIDDLE  ALT=\"46%2F%28x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Putting this altogether, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E3+-+6x%5E2+%2B+8x+-+18%2B46%2F%28x%2B3%29 ALIGN=MIDDLE  ALT=\"2x%5E3+-+6x%5E2+%2B+8x+-+18%2B46%2F%28x%2B3%29\"   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+-+10x%5E2+%2B+6x+-+8%29%2F%28x%2B3%29=2x%5E3+-+6x%5E2+%2B+8x+-+18%2B46%2F%28x%2B3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which looks like this in remainder form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E4+-+10x%5E2+%2B+6x+-+8%29%2F%28x%2B3%29=2x%5E3+-+6x%5E2+%2B+8x+-+18 ALIGN=MIDDLE  ALT=\"%282x%5E4+-+10x%5E2+%2B+6x+-+8%29%2F%28x%2B3%29=2x%5E3+-+6x%5E2+%2B+8x+-+18\"   DISABLED_style= \"max-width:90%; height:auto;\" > remainder 46<BR>\n" );
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