If a,b,c are distinct, and 111abca3b3c3 =(b−c)(c−a)(a−b)(a+b+c) then Δ=111(x−a)2(x−b)2(x−c)2(x−b)(x−c)(x−c)(x−a)(x−a)(x−b) vanishes if

Multiplying C1 by (x-a), C2 by (x-b) and C3 by (x-c), we getΔ=1abcabca3b3c3abcabcabcWhere A=x−a,B=x−b,C=x−cΔ=abca3b3c3111=(−1)(−1)111abca3b3c3                  =(b−c)(c−a)(a−b)(a+b+c)                  =(c−b)(a−b)(b−a)[3x−(a+b+c)]