Let $F = \mathbb{F}_2$ be the field with two elements. I will denote the rings of polynomials and formal power series over $F$ as $F[t]$ and $F[[t]]$ respectively. Suppose that $x \in F[[t]]$ is algebraic over $F[t]$ (there exists a non-zero polynomial $P$ with coefficients in $F[t]$ such that $P(x) = 0$). Is it true that there exists a non-zero polynomial $Q(y) = \sum\limits_{k = 0}^m q_k y^{2^k}$ with coefficients $q_k$ from $F[t]$ such that $Q(x) = 0$ (the difference with is that all non-zero coefficients in $Q$ correspond to the powers of two)?

(I encountered this statement being used without proof in literature, so it most probably is true.)