How to show using factorTheorem,show that a-b,b-c and c-a are the factors of a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)?

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You need to notice that the differences of squares may be converted into products such that:

`a(b -c)(b + c) + b(c - a)(c + a) + c(a - b)(a + b)`

Notice that the factor theorem states that (`x - a` ) is a factor of a polynomial f(x) if and only if `f(a) = 0`  and since the problem does not provide a polynomial, the factor theorem is useless in this case.

Instead, you may openthe brackets such that:

`ab^2 - ac^2 + bc^2 - ba^2 + ca^2 - cb^2`

You may group the terms such that:

`(ab^2 - ba^2) + (ca^2 - ac^2) + (bc^2 - cb^2)`

Factoring out `ab` ,`ac`  and `bc`  yields:

`ab(a - b) + ac(a - c) + bc(b - c)`

Notice the presence of partial factors a - b , a - c , b - c, but they are not the factors of entire given expression  `a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2).`

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