Question

Find the factors of $(b^3+c^3)(b-c)+(c^3+a^3)(c-a)+(a^3+b^3)(a-b)$.

Answer 1

Denote the expression by $E$; then $E$ is a function of $a$ which vanishes when $a=b$, and therefore contains $a-b$ as a factor [Art. 514]. Similarly it contains the factors $b-c$ and $c-a$; thus $E$ contains $(b-c)(c-a)(a-b)$ as a factor.
Also since $E$ is of the fourth degree the remaining factor must be of the first degree; and since it is a symmetrical function of $a$, $b$, $c$, it must be of the form $M(a+b+c)$.
$\therefore E=M(b-c)(c-a)(a-b)(a+b+c)$.
To obtain $M$ we may give to $a$, $b$, $c$ any values that we find most convenient; thus by putting $a=0$, $b=1$, $c=2$, we find $M=1$, and we have the required result.

The required factors are $(a-b)(b-c)(c-a)(a+b+c)$.