Formation of polymers - addition polymerisation In addition polymerisation, a double bond in the monomer is broken and a long chain of polymers is formed. How is this bond broken (where is the energy input) and how is the long chain formed? 

(something to do with free bonds?)
During radical polymerization, we use molecules called initiators $I$ for forming free radicals. The molecules of initiators contain  at least one fragile bond that can easily dissociate upon heating, and yield free radicals.
$$I\rightarrow2A^.$$
The as-formed radicals  have enough energy to attack the monomer molecule $M$ and transfer the radical to the monomer molecule:$$A^.+ M\rightarrow  AM^.$$
More precisely, one electron of the double bond in the monomer forms a covalent bond with the free electron of the radical $A^.$. The other electron of the double bond is now free. We have  a radical  on the monomer  $AM^.$. 

The radical monomer  attacks another monomer molecule, and the polymerization reaction propagates:
$$  AM^. + M \rightarrow AMM^.  $$
In the above reaction, one electron of the double bond in the monomer molecule forms a covalent bond with the free electron of the radical $AM^.$.The other electron of the double bond is now free $AMM^.  $. 
and so on..