[" Hes and for Prove that: "],[[1],[" a "]],[[" 1"," b "+c],[" 1",c+a],[" c ",a+b]]

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

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

Using the property of determinants and without expanding, prove that: |[1,b c, a(b+c)],[1,c a, b(c+a)],[1,a b, c(a+b)]|=0

prove |[1, b c, a(b+c)],[ 1, c a, b(c+a)],[ 1, a b, c(a+b)]|=0

Prove the following : [[1,bc,a(b+c)],[1,ca,b(c+a)],[1,ab,c(a+b)]] =0

Without expanding, prove that : |{:(1, bc, a(b+ c) ),(1, ca, b ( c+ a) ),(1, ab , c( a+ b)):}|=0 .

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

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

prove that det[[1,a,b+c1,b,c+a1,c,a+b]]=0

Prove that |(1, a, a^3),(1, b, b^3),(1, c, c^3)| = (a-b)(b-c)(c-a)(a+b+c).