document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=119007>Question 119007</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve using synthetic division\r<BR>\n" );
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document.write( "(x^4+6x³+6x²)÷(x+5) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #87120 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=87120\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "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%5E4+%2B+6x%5E3+%2B+6x%5E2%29%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x%5E4+%2B+6x%5E3+%2B+6x%5E2%29%2F%28x%2B5%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%2B5=0 ALIGN=MIDDLE  ALT=\"x%2B5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B5 ALIGN=MIDDLE  ALT=\"x%2B5\"   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=-5 ALIGN=MIDDLE  ALT=\"x=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for x.\r<BR>\n" );
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document.write( "so our test zero is -5\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=6x%5E2 ALIGN=MIDDLE  ALT=\"6x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x%5E0 ALIGN=MIDDLE  ALT=\"0x%5E0\"   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%5E1 ALIGN=MIDDLE  ALT=\"x%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is simply because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4+%2B+6x%5E3+%2B+6x%5E2 ALIGN=MIDDLE  ALT=\"x%5E4+%2B+6x%5E3+%2B+6x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > really looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E4%2B6x%5E3%2B6x%5E2%2B0x%5E1%2B0x%5E0 ALIGN=MIDDLE  ALT=\"1x%5E4%2B6x%5E3%2B6x%5E2%2B0x%5E1%2B0x%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</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 1)\r<BR>\n" );
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document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</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 >1</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -5 by 1 and place the product (which is -5)  right underneath the second  coefficient (which is 6)\r<BR>\n" );
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document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</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></TR>\n" );
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document.write( "    Add -5 and 6 to get 1. Place the sum right underneath -5.\r<BR>\n" );
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document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -5 by 1 and place the product (which is -5)  right underneath the third  coefficient (which is 6)\r<BR>\n" );
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document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-5</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >1</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -5 and 6 to get 1. Place the sum right underneath -5.\r<BR>\n" );
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document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-5</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >1</TD><TD >1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -5 by 1 and place the product (which is -5)  right underneath the fourth  coefficient (which is 0)\r<BR>\n" );
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document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-5</TD><TD >-5</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >1</TD><TD >1</TD><TD ></TD><TD ></TD></TR>\n" );
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document.write( "    Add -5 and 0 to get -5. Place the sum right underneath -5.\r<BR>\n" );
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document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-5</TD><TD >-5</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >-5</TD><TD ></TD></TR>\n" );
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document.write( "    Multiply -5 by -5 and place the product (which is 25)  right underneath the fifth  coefficient (which is 0)\r<BR>\n" );
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document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-5</TD><TD >-5</TD><TD >25</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >-5</TD><TD ></TD></TR>\n" );
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document.write( "    Add 25 and 0 to get 25. Place the sum right underneath 25.\r<BR>\n" );
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document.write( "<TR><TD >-5</TD><TD >|</TD><TD >1</TD><TD >6</TD><TD >6</TD><TD >0</TD><TD >0</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >-5</TD><TD >-5</TD><TD >-5</TD><TD >25</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >1</TD><TD >1</TD><TD >-5</TD><TD >25</TD></TR>\n" );
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document.write( "Since the last column adds to 25, we have a remainder of 25. This means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B5 ALIGN=MIDDLE  ALT=\"x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <b>not</b> a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4+%2B+6x%5E3+%2B+6x%5E2 ALIGN=MIDDLE  ALT=\"x%5E4+%2B+6x%5E3+%2B+6x%5E2\"   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 (1,1,1,-5) form the quotient\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+%2B+x%5E2+%2B+x+-+5 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+x%5E2+%2B+x+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and the last coefficient 25, is the remainder, which is placed over <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B5 ALIGN=MIDDLE  ALT=\"x%2B5\"   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=25%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"25%2F%28x%2B5%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=x%5E3+%2B+x%5E2+%2B+x+-+5%2B25%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+x%5E2+%2B+x+-+5%2B25%2F%28x%2B5%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=%28x%5E4+%2B+6x%5E3+%2B+6x%5E2%29%2F%28x%2B5%29=x%5E3+%2B+x%5E2+%2B+x+-+5%2B25%2F%28x%2B5%29 ALIGN=MIDDLE  ALT=\"%28x%5E4+%2B+6x%5E3+%2B+6x%5E2%29%2F%28x%2B5%29=x%5E3+%2B+x%5E2+%2B+x+-+5%2B25%2F%28x%2B5%29\"   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=%28x%5E4+%2B+6x%5E3+%2B+6x%5E2%29%2F%28x%2B5%29=x%5E3+%2B+x%5E2+%2B+x+-+5 ALIGN=MIDDLE  ALT=\"%28x%5E4+%2B+6x%5E3+%2B+6x%5E2%29%2F%28x%2B5%29=x%5E3+%2B+x%5E2+%2B+x+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > remainder 25\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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