Correct power factor formua
I would say that the most general formula is:

$$P=Re\{\underline{S}\}=Re\{\underline{V}·\underline{I}^{\star}\}=S\cos{\theta}$$

where I'm using underline to identify phasors. 

S is the module of \$\underline{S}\$ and it is always positive (it is simply the voltage amplitude times the current amplitude). 

\$\theta\$ is the angle between voltage and current. If this angle is bigger than 90º it means that the load is in reality providing active power to the source and not the other way around. 

Therefore P can be positive or negative.