According to factor theorem, if #f(a)=0#, then #(x–a)# is a factor of the polynomial #f(x)#. Converse of this theorem is also true i.e. if #(x-a)# is a factor of the polynomial #f(x)#, then #f(a)=0#.
Further according to rational zeros theorem, If #f(x)# is a polynomial with integer coefficients and if #p/q# is a zero of #f(x)# i.e. #f(p/q)=0#, then #p# is a factor of the constant term of #f(x)# and #q# is a factor of the leading coefficient of #f(x)#.
Here we have #f(x)=2x^4-3x^3-3x^2+6x-2# and #1/2# and #1# are two zeros of #f(x)#. Therefore two factors of #f(x)# are #2x-1# and #x-1# i.e. #(2x-1)(x-1)=2x^2-3x+1# is a factor.
Dividing #2x^4-3x^3-3x^2+6x-2# by #2x^2-3x+1#, we get #x^2-2=(x+sqrt2)(x-sqrt2)# and we have irrational zeros #sqrt2# and #-sqrt2#.
