Factorise
b^{2}c+c^{2}a+a^{2}b-bc^{2}-ca^{2}-ab^{2 }
Deduce from the factors that changing the values of a, b and c to a+x, b+x and c+x respectively, does not alter the value of the expression.
b^2 c+c^2 a+a^2 b-bc^2-ca^2-ab^2
= 〖( b〗^2 c -ab^2)+〖(c〗^2 a- ca^2)+(a^2 b-bc^2)
= 〖 b〗^2 (c -a)+ac( c- a)+b(a^2-c^2)
= 〖 b〗^2 (c -a)+ac( c- a)- b(c^2- a^2)
= 〖 b〗^2 (c -a)+ac( c- a)- b(c+ a)(c-a)
=( c-a) [〖 b〗^2 +ac- b(c+ a)]
=( c-a) [〖 b〗^2 +ac- bc-ba]
= ( c-a) [〖 (b〗^2 -ba)+(ac- bc)]
=( c-a) [b(b-a)-c(b- a)]
= ( c-a) (b-a)(b-c)