Prove that ` |[1,a,a^3],[1,b,b^3],[1,c,c^3]|=(a-b)(b-c)(c-a)(a+b+c)`

By using properties of determinants. Show that: (i) |[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|=(a-b)(b-c)(c-a) (ii) |[1, 1, 1],[a, b, c],[ a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a+b+c)

Prove: |[1,a^2+bc, a^3],[ 1,b^2+c a, b^3],[ 1,c^2+a b, c^3]|=-(a-b)(b-c)(c-a)(a^2+b^2+c^2)

Show that |[a ,b ,c],[ a^2,b^2,c^2],[bc, ca, ab]|=|[1, 1, 1],[a^2,b^2,c^2],[a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a b+b c+c a) .

Using properties of determinants. Prove that |[3a,-a+b,-a+c],[-b+a,3b,-b+c],[-c+a,-c+b,3c]|=3(a+b+c)(a b+b c+c a)

Prove that |[1+a,1, 1], [1,1+b,1], [1, 1, 1+c]|=a b c(1+1/a+1/b+1/c)=a b c+b c+c a+a b

Prove that |[b+c ,a-b, a] ,[c+a, b-c ,b],[ a+b, c-a, c]|=3abc-a^3-b^3-c^3dot

Prove that =|1 1 1a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

Prove: |a^3 2a b^3 2b c^3 2c|=2(a-b)(b-c)(c-a(a+b+c)

Prove that a^3+b^3+c^3-3abc=1/2(a+b+c){(a-b)^2+(b-c)^2+(c-a)^2}

Using properties of determinants. Prove that |[3a,-a+b,-a+c], [-b+a,3b,-b+c], [-c+a,-c+b,3c]|=3(a+b+c)(a b+b c+c a)