Let $G$ be a connected graph of order $n$ with $n\geq25$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. Nikiforov introduced the $A_α$-matrix of $G$ as $A_α(G)=αD(G)+(1-α)A(G)$ [V. Nikiforov, Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107], where $α\in[0,1]$, $D(G)$ denotes the diagonal matrix of vertex degrees of $G$ and $A(G)$ denotes the adjacency matrix of $G$. The largest eigenvalue of $A_α(G)$, denoted by $λ_α(G)$, is called the $A_α$-spectral radius of $G$. In this paper, it is proved that $G$ has a $\{P_3,P_4,P_5\}$-factor unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $λ_α(G)\geqλ_α(K_1\vee(K_{n-2}\cup K_1))$, where $α$ be a real number with $0\leqα<\frac{2}{3}$.