Given a string $s$ of length $n$ over a general alphabet and an integer $k$, the problem is to decide whether $s$ is a concatenation of $k$ nonempty palindromes. Two previously known solutions for this problem work in time $O(kn)$ and $O(n\log n)$ respectively. Here we settle the complexity of this problem in the word-RAM model, presenting an $O(n)$-time online deciding algorithm. The algorithm simultaneously finds the minimum odd number of factors and the minimum even number of factors in a factorization of a string into nonempty palindromes. We also demonstrate how to get an explicit factorization of $s$ into $k$ palindromes with an $O(n)$-time offline postprocessing.