document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=191251>Question 191251</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780395977255/algebrahomeworkh>algebra and trigonometry structure and method book 2</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of times r is a root of P(x)=0.<BR>\n" );
document.write( "P(x)= x^4+4x^3-16x-16; r=-2\r<BR>\n" );
document.write( "\n" );
document.write( "i found the depressed eruation twice but the numbers were all messed up. please help and explain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #143544 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=143544\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, perform synthetic division where -2 is the test zero (let me know if you need help with synthetic division)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "-2 | 1   4   0   -16   -16\r\n" );
document.write( "   |    -2  -4     8    16\r\n" );
document.write( "   ------------------------\r\n" );
document.write( "     1   2  -4    -8     0\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the last number in the bottom row is zero, this means that the remainder is 0. So -2 is a root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29=+x%5E4%2B4x%5E3-16x-16 ALIGN=MIDDLE  ALT=\"P%28x%29=+x%5E4%2B4x%5E3-16x-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first 4 numbers form the depressed polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B2x%5E2-4x-8 ALIGN=MIDDLE  ALT=\"x%5E3%2B2x%5E2-4x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B4x%5E3-16x-16=%28x%2B2%29%28x%5E3%2B2x%5E2-4x-8%29 ALIGN=MIDDLE  ALT=\"x%5E4%2B4x%5E3-16x-16=%28x%2B2%29%28x%5E3%2B2x%5E2-4x-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now perform synthetic division on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B2x%5E2-4x-8 ALIGN=MIDDLE  ALT=\"x%5E3%2B2x%5E2-4x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the same test zero:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "-2 | 1   2  -4    -8   \r\n" );
document.write( "   |    -2   0     8   \r\n" );
document.write( "   ------------------\r\n" );
document.write( "     1   0  -4     0   \r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how the last number in the bottom row is 0. So -2 is a root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B2x%5E2-4x-8 ALIGN=MIDDLE  ALT=\"x%5E3%2B2x%5E2-4x-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So far, r=-2 is a root of multiplicity 2 (ie -2 is a root twice).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first 3 numbers in the bottom row form the new polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4 ALIGN=MIDDLE  ALT=\"x%5E2-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This tells us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2B2x%5E2-4x-8=%28x%2B2%29%28x%5E2-4%29 ALIGN=MIDDLE  ALT=\"x%5E3%2B2x%5E2-4x-8=%28x%2B2%29%28x%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B4x%5E3-16x-16=%28x%2B2%29%28x%5E3%2B2x%5E2-4x-8%29=%28x%2B2%29%28x%2B2%29%28x%5E2-4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now perform synthetic division on the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4 ALIGN=MIDDLE  ALT=\"x%5E2-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "-2 | 1   0  -4     \r\n" );
document.write( "   |    -2   4    \r\n" );
document.write( "   ------------\r\n" );
document.write( "     1  -2   0\r\n" );
document.write( "\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So -2 is a root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4 ALIGN=MIDDLE  ALT=\"x%5E2-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means that r=-2 is a root of multiplicity 3 (ie -2 is a root three times).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first two numbers in the bottom row form the new polynomial: <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;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now because -2 is NOT a root of <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;\" >, this means that we can stop looking for more roots of -2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B4x%5E3-16x-16=%28x%2B2%29%28x%5E3%2B2x%5E2-4x-8%29=%28x%2B2%29%28x%2B2%29%28x%5E2-4%29=%28x%2B2%29%28x%2B2%29%28x%2B2%29%28x-2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Or in other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B4x%5E3-16x-16=%28x%2B2%29%28x%2B2%29%28x%2B2%29%28x-2%29 ALIGN=MIDDLE  ALT=\"x%5E4%2B4x%5E3-16x-16=%28x%2B2%29%28x%2B2%29%28x%2B2%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how the factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2 ALIGN=MIDDLE  ALT=\"x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is repeated 3 times, this supports our conclusion that r=-2 is a root of multiplicity 3<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );