Without expanding show that : `|(a,b+c,1),(b,c+a,1),(c,a+b,1)|=0`

Without expanding show that : |(a+b,b+c,c+a),(b+c,c+a,a+b),(c+a,a+b,b+c)|=2|(a,b,c),(b,c,a),(c,a,b)|

Without expanding show that : |(bc,a,a^2),(ca,b,b^2),(ab,c,c^2)|=|(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3)|

Without expanding show that : |(1,a,a^2-bc),(1,b,b^2-ca),(1,c,c^2-ab)|=0

Without expanding show that : |(1,bc,bc(b+c)),(1,ca,ca(c+a)),(1,ab,ab(a+b))|=0

Without expanding show that : |(1,a,a^2),(1,b,b^2),(1,c,c^2)|=|(1,bc,b+c),(1,ca,c+a),(1,ab,a+b)|

Using properties of determinant show that : |(a,b+c,a^2),(b,c+a,b^2),(c,a+b,c^2)|=-(a+b+c)(a-b)(b-c)(c-a)

Using properties of determinant show that : |(a+b+2c,a,b),(c,b+c+2a,b),(c,a,c+a+2b)|=2(a+b+c)^3

Using properties of determinant show that : |(1+a,1,1),(1,1+b,1),(1,1,1+c)|=abc+ab+bc+ca

Show that: (a-b)(a+b)+ (b-c) (b+c) +(c-a) (c+a) =0

Using properties of determinant show that : |(a,b,c),(a^2,b^2,c^2),(b+c,c+a,a+b)|=(a+b+c)(a-b)(b-c)(c-a)