The Waring Problem for Matrix Algebras, II

Let $f$ bea noncommutativepolynomial of degree $m\ge 1$ over an algebraically closed field $F$ of characteristic $0$. If $n\ge m-1$ and ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are nonzero elements from $F$ such that ${\alpha }_1+{\alpha }_2+{\alpha }_3=0$, then every trace zero $n\times n$ matrix over $F$ can be written as ${\alpha }_1{A}_1+{\alpha }_2{A}_2+{\alpha }_3{A}_3$ for some $A_i$ in the image of $f$ in $M_n(F)$.