Answer
At point $P$, $~~~V = 0$
Work Step by Step
We can write a general equation for the net electric potential at a point:
$V_{net} = \frac{1}{4\pi~\epsilon_0}\sum \frac{q_i}{r_i}$
By symmetry, the electric potential due to the positive charge on the left half of the rod cancels out with the electric potential due to the negative charge on the right half of the rod.
At point $P$, $~~~V = 0$.