Is this polynomial in three variables irreducible?

Question

The polynomial is $x_0^3 + x_1^3 + x_2^3 - (x_0 + x_1 + x_2)^3$ over an algebraically closed field of nonzero characteristic. I tried by writing the trinomial cube more explicitly, but that didn’t help. Any hints? Also, how about a very similar polynomial $x_0^3 + x_1^3 + x_2^3 - \frac{(x_0 + x_1 + x_2)^3}{9}$?

Answer 1

$$x_0^3 + x_1^3 + x_2^3 - (x_0 + x_1 + x_2)^3= [x_0^3 + x_1^3] + [x_2^3 - (x_0 + x_1 + x_2)^3].$$

Both summands are divisible by $x_0+x_1$ so the polynomial is reducible over any field. It is in fact $0$ iff the characteristic is $3$.

Answer 2

$$(a+b+c)^3-a^3-b^3-c^3=\sum_{cyc}(3a^2b+3a^2c+2abc)=3(a+b)(a+c)(b+c).$$