Prove that (a+b+c+x) is a factor of `|(x+a, b,c),(b,x+c,a),(c,a,x+b)|`

One root of the equation |(x+a,b,c),(b,x+c,a),(c,a,x+b)|=0 is -

Solve : |(x + a, b, c),(a, x+b, c),(a, b, x + c)| =0

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

If x+y+z=0 prove that |[a x, b y, c z],[ c y, a z ,b x],[ b z ,c x ,a y]|=x y z|[a, b, c],[c ,a ,b],[b ,c ,a]|

Prove that both the roots of the equation (x-a)(x-b)+ (x-b)(x-c)+(x-a)(x-c)=0 are real.

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is (a) |[a, -c, -b], [-c, b, -a], [-b, -a, c]| (b) |[a, b, c], [b, c, a], [c, a, b]| (c) |[a, c, b], [b, a, c], [c, b, a]| (d) |[a, a+b, b+c], [b, b+c, c+a], [c, c+a, a+b]|

If f(x)=a+b x+c x^2 and alpha,beta,gamma are the roots of the equation x^3=1,t h e n \ |[a,b,c],[b,c,a],[c,a,b]| is equal to

|{:(a-x," "b," "c),(" "b,c-x," "a),(" "c," "a,b-x):}|=0

|{:(x+a," "b," "c),(" "a,x+b," "c),(" "a," "b,x+c):}|=0

If alpha and beta be the roots of the equation (x-a)(x-b)=c ( cne0 ),prove that a and b are the roots of the equation (x-alpha)(x-beta)+c=0 .