|[1,x,x^(3)],[1,b,b^(3)],[1,c,c^(3)]|=0

det[[1,x,x^(3)1,b,b^(3)1,c,c^(3)]]=0;b!=c

If [[a,a^(2),a^(3)-1b,b^(2),b^(3)-1c,c^(2),c^(3)-1]]=0

If |(1,1,1),(a,b,c),(a^(3),b^(3),c^(3))| = (a - b) (b - c) (c - a) (a + b + c) , where a,b,c are all different, then the determinant |(1,1,1),((x-a)^(2),(x-b)^(2),(x-c)^(2)),((x-b)(x-c),(x-c)(x-a),(x-a)(x-b))| vanishes when a)a + b + c = 0 b) x = (1)/(3) (a + b + c) c) x = (1)/(2) (a + b + c) d) x = a + b + c

If |(1,1,1),(a,b,c),(a^(3),b^(3),c^(3))|= (a -b) (b -c) (c -a) (a + b+c) where a, b, c are all different, then the determinant |(1,1,1),((x-a)^(2),(x -b)^(2),(x -c)^(2)),((x -b) (x -c),(x -c) (x -a),(x -a) (x -b))| vanishes when

If |(1,1,1),(a,b,c),(a^(3),b^(3),c^(3))|= (a -b) (b -c) (c -a) (a + b+c) where a, b, c are all different, then the determinant |(1,1,1),((x-a)^(2),(x -b)^(2),(x -c)^(2)),((x -b) (x -c),(x -c) (x -a),(x -a) (x -b))| vanishes when

if |{:(1,,1,,1),(a,,b,,c),(a^(3),,b^(3),,c^(3)):}|= (a-b)(b-c)(c-a)(a+b+c) where a,b,c are all different then the determinant |{:(1,,1,,1),((x-a)^(2),,(x-b)^(2),,(x-c)^(2)),((x-b)(x-c),,(x-c)(x-a),,(x-a)(x-b)):}| vanishes when