document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92192>Question 92192</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0395977258/algebrahomeworkh>Algebra and Trigonometry</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> please me find the answer by useing  the Factor Theorem to determine whether<BR>\n" );
document.write( "(x- 3) is a factor of  f(x) = x^4 + 12x^3 + 6x + 27  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #67143 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=67143\">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=%28x%5E4+%2B+12x%5E3+%2B+6x+%2B+27%29%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"%28x%5E4+%2B+12x%5E3+%2B+6x+%2B+27%29%2F%28x-3%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-3=0 ALIGN=MIDDLE  ALT=\"x-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the denominator <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;\" > 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 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=12x%5E3 ALIGN=MIDDLE  ALT=\"12x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E1 ALIGN=MIDDLE  ALT=\"6x%5E1\"   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%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   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+12x%5E3+%2B+6x+%2B+27 ALIGN=MIDDLE  ALT=\"x%5E4+%2B+12x%5E3+%2B+6x+%2B+27\"   DISABLED_style= \"max-width:90%; height:auto;\" > really looks like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E4%2B12x%5E3%2B0x%5E2%2B6x%5E1%2B27x%5E0 ALIGN=MIDDLE  ALT=\"1x%5E4%2B12x%5E3%2B0x%5E2%2B6x%5E1%2B27x%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</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" );
document.write( "</TABLE>\r<BR>\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 >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</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" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Multiply 3 by 1 and place the product (which is 3)  right underneath the second  coefficient (which is 12)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</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" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Add 3 and 12 to get 15. Place the sum right underneath 3.\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD ></TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >15</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Multiply 3 by 15 and place the product (which is 45)  right underneath the third  coefficient (which is 0)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >45</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >15</TD><TD ></TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Add 45 and 0 to get 45. Place the sum right underneath 45.\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >45</TD><TD ></TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >15</TD><TD >45</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Multiply 3 by 45 and place the product (which is 135)  right underneath the fourth  coefficient (which is 6)\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >45</TD><TD >135</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >15</TD><TD >45</TD><TD ></TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Add 135 and 6 to get 141. Place the sum right underneath 135.\r<BR>\n" );
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document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >45</TD><TD >135</TD><TD ></TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >15</TD><TD >45</TD><TD >141</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Multiply 3 by 141 and place the product (which is 423)  right underneath the fifth  coefficient (which is 27)\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >45</TD><TD >135</TD><TD >423</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >15</TD><TD >45</TD><TD >141</TD><TD ></TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "    Add 423 and 27 to get 450. Place the sum right underneath 423.\r<BR>\n" );
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document.write( "    <TABLE  cellpadding=10>\n" );
document.write( "<TR><TD >3</TD><TD >|</TD><TD >1</TD><TD >12</TD><TD >0</TD><TD >6</TD><TD >27</TD></TR><TR><TD ></TD><TD >|</TD><TD ></TD><TD >3</TD><TD >45</TD><TD >135</TD><TD >423</TD><TD ></TD></TR><TR><TD ></TD><TD ></TD><TD >1</TD><TD >15</TD><TD >45</TD><TD >141</TD><TD >450</TD></TR>\n" );
document.write( "</TABLE>\r<BR>\n" );
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document.write( "Since the last column adds to 450, we have a remainder of 450. This means <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;\" > is <b>not</b> a factor of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4+%2B+12x%5E3+%2B+6x+%2B+27 ALIGN=MIDDLE  ALT=\"x%5E4+%2B+12x%5E3+%2B+6x+%2B+27\"   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 (1,15,45,141) 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+15x%5E2+%2B+45x+%2B+141 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+15x%5E2+%2B+45x+%2B+141\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and the last coefficient 450, is the remainder, which is placed over <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;\" > like this\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=450%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"450%2F%28x-3%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+15x%5E2+%2B+45x+%2B+141%2B450%2F%28x-3%29 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+15x%5E2+%2B+45x+%2B+141%2B450%2F%28x-3%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+12x%5E3+%2B+6x+%2B+27%29%2F%28x-3%29=x%5E3+%2B+15x%5E2+%2B+45x+%2B+141%2B450%2F%28x-3%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=%28x%5E4+%2B+12x%5E3+%2B+6x+%2B+27%29%2F%28x-3%29=x%5E3+%2B+15x%5E2+%2B+45x+%2B+141 ALIGN=MIDDLE  ALT=\"%28x%5E4+%2B+12x%5E3+%2B+6x+%2B+27%29%2F%28x-3%29=x%5E3+%2B+15x%5E2+%2B+45x+%2B+141\"   DISABLED_style= \"max-width:90%; height:auto;\" > remainder 450\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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