document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=192382>Question 192382</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find exact values for all real/complex number solutions to the polynomial equation...\r<BR>\n" );
document.write( "\n" );
document.write( "3x^4 - 5x^3 + 5x^2 - 5x + 2 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Could you please provide me with a very detailed explanation of how you achieved the solution. Thanks! <BR>\n" );
document.write( "        </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #144439 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=144439\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> techniques  that  apply  include:  rational  zero  theorem, synthetic  division,  and  fundamental  theorem  of  algebra.\r<BR>\n" );
document.write( "\n" );
document.write( "Fundamental  theorem  states  that  a  4th  degree  function  has  4  roots.\r<BR>\n" );
document.write( "\n" );
document.write( "rational  zero  theorem  states  that  zeros  are  ratios  of factors  of last  term  coeff  and  leading  coeff\r<BR>\n" );
document.write( "\n" );
document.write( "coeff  are   2,3  therefore  factors  are  +/-  ( 2/1, 1/3, 1/1, 2/3)\r<BR>\n" );
document.write( "\n" );
document.write( "using  synthetic  div  first\r<BR>\n" );
document.write( "\n" );
document.write( "1          3  -5  5  -5  2<BR>\n" );
document.write( "                 3   -2  3  -2<BR>\n" );
document.write( "           _________________<BR>\n" );
document.write( "             3  -2   3  -2  0            \r<BR>\n" );
document.write( "\n" );
document.write( "therefore  (x-1)   is  a  factor,  x=1  is  ZERO\r<BR>\n" );
document.write( "\n" );
document.write( "remaining  poly  is   3x^3-2x^2+3x-2\r<BR>\n" );
document.write( "\n" );
document.write( "try  factoring  by  grouping,  rearranging,\r<BR>\n" );
document.write( "\n" );
document.write( "3x^3 +3x  -2x^2 -2\r<BR>\n" );
document.write( "\n" );
document.write( "3x(x^2+1) -2(x^2+1)\r<BR>\n" );
document.write( "\n" );
document.write( "(3x-2) (x^2+1)\r<BR>\n" );
document.write( "\n" );
document.write( "setting  1st  term  =0\r<BR>\n" );
document.write( "\n" );
document.write( "3x-2=0  <BR>\n" );
document.write( "3x=2<BR>\n" );
document.write( "x=2/3          ZERO\r<BR>\n" );
document.write( "\n" );
document.write( "2nd  term,  (x^2+1)  is  not  readily  factored  but  quadratic  eqn  works\r<BR>\n" );
document.write( "\n" );
document.write( "using  a=1,  b=0,  c=1\r<BR>\n" );
document.write( "\n" );
document.write( "x=  ( -0 +/- sq  rt  ( 0^2 -(4) (1) (1) ) / 2 *1\r<BR>\n" );
document.write( "\n" );
document.write( "x= ( +/-  sq  rt  (  -4 )  ) /2\r<BR>\n" );
document.write( "\n" );
document.write( "x=+/- 2i/2=+/- (i),    LAST  2  ZEROS\r<BR>\n" );
document.write( "\n" );
document.write( " where  i= sq  rt  (-1)    \r<BR>\n" );
document.write( "\n" );
document.write( "in  summary,  zeros  are  (1),  (2/3),  (+i),  (-i)\r<BR>\n" );
document.write( "\n" );
document.write( "check  by  multiplying  factors\r<BR>\n" );
document.write( "\n" );
document.write( "(x^2+1) (x-1) (3x-2)<BR>\n" );
document.write( "(x^2+1) (3x^2 -5x+2)<BR>\n" );
document.write( "3x^4 -5x^3 +2x^2 +3x^2 -5x+2<BR>\n" );
document.write( "3x^4-5x^3+5x^2-5x+2      ok </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );