For a sequence of polynomials $P_n(x):=\sum_{m\le n}p_mx^m$, $n\ge 1$, we give a necessary and sufficient condition for the asymptotic equivalence $$ P_n^{(\alpha)}(x):=\sum_{młe n}c_mp_mx^m\sim c_nP_n(x) \quad (n\to\infty), $$ to hold for each $x\ge A$ and an arbitrary regularly varying sequence $\{c_n\}$ of index $\alpha\in R$.