How to factor a polynomial expression in MATHEMATICA?

Question

I want to factorize the following polynomial in MATHEMATICA: $1 - 2 r + r^2 - 2 s + 2 r s + s^2 - 2 t + 2 r t + 2 s t - 4 r s t + t^2$.

1 - 2 r + r^2 - 2 s + 2 r s + s^2 - 2 t + 2 r t + 2 s t - 4 r s t + t^2

If done by hand it is easy to see that the above expression can be written in the form of (a+b)(a-b) as: $(-1 + r + s + t)^2 - (2 \sqrt{r s t})^2$.

I tried using the "Factor" command in MATHEMATICA but it doesn't help, so if there is any command or any code that can help me with the same is appreciated.

Edit: This specific factorization is required as I want the degree of each factored part to be 1. And also if there is any general algorithm so as to obtain factorization for such polynomials in $(a-b)(a+b)$ form so that it can be used with other such polynomials too.

Answer 1

Perhaps this is what you can use in this case

ex = 1-2r+r^2-2s+2r s+s^2-2t+2r t+2s t-4r s t+t^2;
(ex/.r s t->u^2//Factor)/.u->Sqrt[r s t]

which returns

(-1 + r + s + t - 2*Sqrt[r*s*t])*(-1 + r + s + t + 2*Sqrt[r*s*t]

I don't think this can be generalized. Any expression with more than one term can be written as $$ a-b = (\sqrt{a})^2 - (\sqrt{b})^2= (\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b})$$ but the choice of $a$ and $b$ is arbitrary but also with the proper choice of signs of the square roots.

Answer 2

Another way assuming $r\ge 0,s\ge 0,t\ge 0$.

expr = 1 - 2 r + r^2 - 2 s + 2 r s + s^2 - 2 t + 2 r t + 2 s t - 4 r s t + t^2;
EXPR = expr /. {s -> S^2 , r -> R^2, t -> T^2} // Factor
EXPR /.{S -> Sqrt[s], R -> Sqrt[r], T -> Sqrt[t]}