If `a , b , c ,` are roots of the equation `x^3+p x+q=0,` prove that `|[a ,b, c],[ b ,c, a],[ c, a, b]|=0`

If a ,\ b ,\ c are roots of the equation x^3+p x+q=0 , prove that |a b c b c a c a b|=0 .

if a,b and c are the roots of the equation x^3+2x^2+1=0 , find |{:(a,b,x),(b,c,a),(c,a,b):}|

The roots of the equation (b-c)x^(2)+(c-a)x+(a-b)=0

If a+b+c=0, solve the equation: |[a-x, c, b],[c,b-x, a],[b, a, c-x]|=0

If a,b,c are in A.P. then the roots of the equation (a+b-c)x^2 + (b-a) x-a=0 are :

If a, b, c are the roots of the equation x^3-12x^2+39x-21=0 then find the value of the determinant |[a, b-c, c+b] , [ a+c, b, c-a] , [a-b, b+a, c]|

If c,d are the roots of the equation (x-a)(x-b)-k=0, prove that a a,b are roots of the equation (x-c)(x-d)+k=0

If the roots of the equation (b-c)x^(2)+(c-a)x+(a-b)=0 are equal,then prove that 2b=a+c