ag.algebraic geometry – reducibility of an element in a linear system in $mathbb{P}^3$

It is known that in characteristic zero, in a linear system (not necessarily complete) in $mathbb{P}^3$ a general element is irreducible. Now my question is the following:
Suppsoe we have a subspace $V subset H^0(mathbb{P}^3, mathcal{O}(3))$ of dimension $ge 3$. Then is it possible always to find an element in $V$ which is reducible ?