In $\mathbb Z[x_1,x_2,\ldots,x_n]$, show $x_1 x_2\cdots x_n$ has $2^{n+1}-2$ non-constant polynomials dividing it This particular question is a part of an assignment which could not be discussed due to pandemic .

Question (a)  Consider the ring R of polynomials in n variables with integer coefficients . Prove that the polynomial $f(x_1 , x_2 ,\ldots, x_n) = x_1 x_2\cdots x_n $ has $2^{n+1}-2$ non-constant polynomials in R dividing it .

Attempt: (a) any polynomial in R dividing f would be of form ${x_{1}}^{i} ...{x_{n}}^{i}$ i=0 or 1.
So , there will be $2^{n}-1$ polynomials . But the answer is 2 times my answer.
Thanks!!
Note that if $x_i ...x_k $ divides f then - ($x_i ...x_k$) also divides f . So, correct answer is $2^{n+1}-2$.
In a) you forgot the coefficient: $-x_1$ divides $x_1x_2$. This gives $2^{n+1}-2$ divisors because the coefficient can be $1$ or $-1$.
b) The phrase &quot;So, the elemets of order 11 is fixed&quot; does not make sense.