Factoring 3 variable Polynomial

Question

Factor $ab^3 - a^3 b + bc^3 - b^3 c + ca^3 - c^3a$.

I have tried to pug in values that would equal zero and possibly make use of the multivariable factor theorem.

Answer 1

$$(a+b+c)( a -b) (b-c)(c-a) $$

parisize = 4000000, primelimit = 500000
? f = -1 * (a+b+c)* ( a -b)* (b-c)*(c-a) 
%1 = (b - c)*a^3 + (-b^3 + c^3)*a + (c*b^3 - c^3*b)
? -f
%2 = (-b + c)*a^3 + (b^3 - c^3)*a + (-c*b^3 + c^3*b)

Factor $ab^3 - a^3 b + bc^3 - b^3 c + ca^3 - c^3a$.

Answer 2

$$ab^3 - a^3 b + bc^3 - b^3 c + ca^3 - c^3a=$$ $$ab(b-a)(b+a)+c^3(b-a)-c(b^3-a^3)=$$ $$=(b-a)(a^2b+b^2a+c^3-b^2c-abc-a^2c)=$$ $$=(b-a)(a^2(b-c)+ab(b-c)-c(b^2-c^2))=$$ $$=(b-a)(b-c)(a^2+ab-bc-c^2)=$$ $$=(b-a)(b-c)((a-c)(a+c)+b(a-c))=$$ $$=(b-a)(b-c)(a-c)(a+b+c).$$