Left Factoring

In general, if $A$ is a single non-terminal, and $\alpha$, $\beta$ and $\gamma$ are strings of terminals or non- terminals, then a production rule of the form: $A\; \rightarrow\; \alpha \beta\; \mid\; \alpha \gamma$ can be left-factored into two rules of the form:

$$\begin{array}{rcl} A & \rightarrow & \alpha\; B \\ B & \rightarrow & \beta\; \mid\; \gamma \\ \end{array}$$

where $B$ is some new non-terminal.