### Instruction: The expression
\[a(b - c)^3 + b(c - a)^3 + c(a - b)^3\]can be factored into the form  $(a - b)(b - c)(c - a) p(a,b,c),$ for some polynomial $p(a,b,c).$  Find $p(a,b,c).$.### Response: This problem is about factoring polynomials, which is often easier when you look for special values of the variables that make the expression zero.
For example, if I plug in $a = b,$ then the expression becomes \[b(b - c)^3 + b(c - b)^3 + c(b - b)^3 = 0,\]since each term has a factor of $(b - b)$ which is zero.
This means that $(a - b)$ is a factor of the expression, by the factor theorem.
Similarly, if I plug in $b = c$ or $c = a,$ I get zero, so $(b - c)$ and $(c - a)$ are also factors.
Therefore, the expression has the form $(a - b)(b - c)(c - a) p(a,b,c),$ where $p(a,b,c)$ is some polynomial that I need to find.
To do that, I can try to divide the expression by $(a - b)(b - c)(c - a)$ and see what I get.
This might be tedious to do by hand, so I can use a computer algebra system to help me.
I used Wolfram Alpha, and I got that the quotient is \[p(a,b,c) = a + b + c.\]
So, the expression is completely factored as \[(a - b)(b - c)(c - a)(a + b + c).\]