PolynomialGCD[6  (x^3) - 17  (x^2) + 14  x - 3, 
      4  (x^4) - 14  (x^3) + 12  (x^2) + 2  x - 3, -8 (x^3) + 14  (x^2) - x - 
        3]

PolynomialGCD[6 (x^3) - 16 (x^2) + ((23 x)/2) - (3/2), 
      4 (x^4) - 14 (x^3) + 12 (x^2) + x - (3/2), -8 (x^3) + 14 (x^2) - 
        2 x - (3/2)]