A graph is (P₅,gem)-free, when it does not contain P₅ (an induced path with five vertices) or a gem (a graph formed by making an universal vertex adjacent to each of the four vertices of the induced path P₄) as an induced subgraph. We present O(n²) time recognition algorithms for chordal gem-free graphs and for (P₅,gem)-free graphs. Using a characterization of (P₅,gem)-free graphs by their prime graphs with respect to modular decomposition and their modular decomposition trees [A. Brandstädt, D. Kratsch, On the structure of (P₅,gem)-free graphs, Discrete Appl. Math. 145 (2005), 155–166], we give linear time algorithms for the following NP-complete problems on (P₅,gem)-free graphs: Minimum Coloring; Maximum Weight Stable Set; Maximum Weight Clique; and Minimum Clique Cover.