Simplify the following circuit and find the boolean polynomial.

Given a circuit

The circuit $pq$ is true when $p$ is true or $q$ is true.

Hence, the circuit $pq$ is represented by $p\vee q$.

Similarly, the circuit $pr$ is true when $p$ is true or $r$ is true.

Hence, the circuit $pr$ is represented by $p\vee r$.

Now, the whole circuit is true only when both $pq$ and $pr$ is true.

Thus, the whole circuit is represented by $\left(p\vee q\right)\wedge \left(p\vee r\right)$.

According to the distributive law $\left(p\vee q\right)\wedge \left(p\vee r\right)=p\vee (q\wedge r)$.