How to use the factor theorem on $a(b^2-c^2) + b(c^2-a^2) + c(a^2-b^2)$?

Question

I know the factor theorem i.e,

Let $P(x)$ be a polynomial of degree greater than or equal to $1$ and $a$ be a real number such that $P(a) = 0$, then $(x-a)$ is a factor of $P(x)$.

I have an question in my textbook which is -

• Using Factor theorem , show that $a-b,b-c$ and $c-a$ are the factors of $$a(b^2 -c^2)+b(c^2-a^2)+c(a^2-b^2).$$

I can not see any polynomial over here . How can I solve the problem ?

Hints are welcome

Answer 1

Instead of a polynomial in $x$ or $y$ as usual, consider the expression $a(b^2 -c^2)+b(c^2-a^2)+c(a^2-b^2)$ to be firstly a polynomial in $a$, then in $b$ then $c$. Each use of the factor theorem on each case should give you one solution.

Answer 2

Hint: Using the difference of two squares we can factorise your expression as $$a(b^2 -c^2)+b(c^2-a^2)+c(a^2-b^2) = a(b-c)(b+c) + b(c-a)(c+a) + c(a-b)(a+b)$$