Show that #a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)# us exactly divisible by #a-b,b-c and c-a#?

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Answer 1 · 2018-06-11T14:28:50

#a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)#

#=a^2b^2(a-b)+b^3c^2-b^2c^3+c^3a^2-c^2a^3#

#=a^2b^2(a-b)-c^2(a^3-b^3)+c^3(a^2-b^2)#

#=(a-b)(a^2b^2-c^2(a^2+ab+b^2)+c^3(a+b))#

#=(a-b)(a^2b^2-c^2a^2-c^2ab-c^2b^2+c^3a+c^3b)#

#=(a-b)(a^2(b^2-c^2)-c^2a(b-c)-c^2b(b-c))#

#=(a-b)(b-c)(a^2b+a^2c-c^2a-c^2b)#

#=(a-b)(b-c)(b(a^2-c^2)+ac(a-c))#

#=(a-b)(b-c)(a-c)(ab+bc+ca)#

This factorisation proves that the given expression is exactly divisible by #(a-b),(b-c) and (c-a)#