document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=192853>Question 192853</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me to solve this polynomial:\r<BR>\n" );
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document.write( "f(x)=(x-8)(x+3)(x-9)(x+8)(x-2)(x+5)(x-2)\r<BR>\n" );
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document.write( "I have to write ALL the process down. I tried it and got a result, but when I graph it, it was different to my original function.\r<BR>\n" );
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document.write( "Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #144777 by </B><A HREF=/tutors/aboutme.mpl?userid=RAY100><B>RAY100(1637)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=RAY100><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=144777\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This  is  an interesting  problem,  if  only  from  its  size.\r<BR>\n" );
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document.write( "we  started  by  arranging  the  factors  in  order.  \r<BR>\n" );
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document.write( "y=(x-9)(x-8)(x-2)(x-2)(x-3)(x-5)(x-8)\r<BR>\n" );
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document.write( "note  this  is  a  7th  degree function.  This  means(7  roots)  and  it  is  an  odd  function\r<BR>\n" );
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document.write( "we can  find  the  roots  from  the  factors,  for  example set  x-9=0,  x=-9  root  likewise  the  roots  in  order  are,  9,8,2,2,3,5,8.\r<BR>\n" );
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document.write( "Note  that  the  root  (2) happens  twice,  this  means  the  function  curve  just  touches  the  y=0  axis  but  does  not  pass  thru.  With  the  other  roots  the  function  intersects  and  passes  thru.\r<BR>\n" );
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document.write( "note  it  is  a  positive  function.  If  we  look  close  the  leading  coefficient  will  be  plus,  because  we  have  no  negative (x)  in  factors.  After  we  multiply  all  factors,  this  holds  true.\r<BR>\n" );
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document.write( "Summarizing,  it is  a  positive  odd  function,  with  7 roots,  and  these  roots  are  defined  by  the  factors. \r<BR>\n" );
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document.write( "The  odd  positive  nature  indicates as  we  graph  the  function  starts at  the  left  with  a  negative  and  moves  upward  to  positive  right.  \r<BR>\n" );
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document.write( "If  we  use  a  standard  x - y  coordinate  system  and  make  just  a  rough  sketch, we  can  define  the  shape  of  the  function.\r<BR>\n" );
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document.write( "starting  at  lower  left  we  rise  thru x=(-8), turn and  pass  downward  thru  x=(-5) turn  and  pass  upward  thru  (x=-3),  turn  downward  and  just  touch  the  axis  at  x=(+2) before  turning  upward  and  then  downward  thru  x=(8),and  then  turn  and  pass upward  thru  x=(9)  and  continuing  upwards.  \r<BR>\n" );
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document.write( "We  see  6  turns  which  is  consistent  with  a  7  degree  function\r<BR>\n" );
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document.write( "We  would  probably  say  the  problem  is  complete  here,  but  you  seemed  to  want  to  expand  the  function.  This  is  a  lot  of  work  and many  chances  for  numerical  error.  \r<BR>\n" );
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document.write( "I  started  with  a  clean  sheet of  paper  sideways.\r<BR>\n" );
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document.write( "i  did  2  factor  by  themselves,  as  this  allowed  the  use  of  foil\r<BR>\n" );
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document.write( "1)(x+8)(x-8)=  x^2-64<BR>\n" );
document.write( "2)(x-2)(x-2)=x^2-4x-4<BR>\n" );
document.write( "3)(x+3)(x-7)=x^2 -4x-21<BR>\n" );
document.write( "4)(x-5)   had  no  mate\r<BR>\n" );
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document.write( "now  we  multiplied 1*2  and  3*4\r<BR>\n" );
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document.write( "5)  x^4-4x^3-60x^2+256x-256<BR>\n" );
document.write( "6)x^3-x^2-57x-135\r<BR>\n" );
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document.write( "and  lastly  we  multiplied  5*6<BR>\n" );
document.write( "but  we  set  each  distribution  on  a  horizontal  line  and  matched  the  next  distribution  below.<BR>\n" );
document.write( "This  resulted  in a  vertical  accounting  of  the  for  factors, one  vertical  column  for  each  degree.\r<BR>\n" );
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document.write( "The  result  was<BR>\n" );
document.write( "y=x^7-5x^6-113x^5+409x^4+3960x^3+9303x^2-19968x+34560\r<BR>\n" );
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document.write( "my  graphing  calc  did  not  like  this  but  i  feel  the  above  sketch  is  correct.\r<BR>\n" );
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document.write( "to  recheck,  i  would  wait  a  day  and  redo\r<BR>\n" );
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