For a multilinear polynomial $f$ of degree $m$ over field $F$, we say that an algebra $A$ is $f$-zpd, if every multilinear functional $\varphi:A^m\rightarrow F$ that preserves zeros of $f$ is of form $\varphi(x_1,\ldots,x_m) = \tau(f(x_1,\ldots,x_m))$ for all $x_1,\ldots,x_m$ in $A$. We will be interested in finding $f$-zpd algebras, focusing particularly on matrix algebras $M_d(F)$. Moreover, we will try to answer a similar problem when a multilinear polinomial $g$ is preserving zeros of a multilinear polynomial $f$ for some algebra. Finally we will consider an application of $f$-zpd algebras on linear maps that preserve zero products, ie. on linear maps $T$ between algebras $A$ and $B$ such that $T(x)T(y)=0$ if $xy=0$.
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