<PRE><B>Question 192382</B>
techniques  that  apply  include:  rational  zero  theorem, synthetic  division,  and  fundamental  theorem  of  algebra.

Fundamental  theorem  states  that  a  4th  degree  function  has  4  roots.

rational  zero  theorem  states  that  zeros  are  ratios  of factors  of last  term  coeff  and  leading  coeff

coeff  are   2,3  therefore  factors  are  +/-  ( 2/1, 1/3, 1/1, 2/3)

using  synthetic  div  first

1          3  -5  5  -5  2
                 3   -2  3  -2
           _________________
             3  -2   3  -2  0            

therefore  (x-1)   is  a  factor,  x=1  is  ZERO

remaining  poly  is   3x^3-2x^2+3x-2

try  factoring  by  grouping,  rearranging,

3x^3 +3x  -2x^2 -2

3x(x^2+1) -2(x^2+1)

(3x-2) (x^2+1)

setting  1st  term  =0

3x-2=0  
3x=2
x=2/3          ZERO

2nd  term,  (x^2+1)  is  not  readily  factored  but  quadratic  eqn  works

using  a=1,  b=0,  c=1

x=  ( -0 +/- sq  rt  ( 0^2 -(4) (1) (1) ) / 2 *1

x= ( +/-  sq  rt  (  -4 )  ) /2

x=+/- 2i/2=+/- (i),    LAST  2  ZEROS

 where  i= sq  rt  (-1)    

in  summary,  zeros  are  (1),  (2/3),  (+i),  (-i)

check  by  multiplying  factors

(x^2+1) (x-1) (3x-2)
(x^2+1) (3x^2 -5x+2)
3x^4 -5x^3 +2x^2 +3x^2 -5x+2
3x^4-5x^3+5x^2-5x+2      ok</PRE>