Can there be a polynomial over the field $F_p$ of $p$ elements ($p$ prime) in non-commuting variables $X_1,..., X_r$ such that:

1) $f(A_1,...,A_r)=0$ for every $n \times n$ matrices $A_1,...,A_r$ over $F_p$; and

2) $f$ has a non-trivial monomial of total degree less than $n$ ?