how do microtubules polymerize?
Microtubules are made up by $\alpha$- and $\beta$-tubulin units, the wiki page for this will link you to a short section on microtubule dynamics, which is the basis of how the growth &amp; shrinking of microtubules is controlled:

Microtubules grow (by addition of $\alpha\beta$-units) and shrink (by dissociation of these units) primarily at the '+ end', which is the one in the cell periphery not at the centriole. Actually both of these reactions (growing/polymerisation and shrinking/de-polymerisation) usually happen at the same time but with different speeds. These speeds depend on whether GDP or GTP is bound by the $\beta$-tubulin at the + end, the (relative) concentration of GTP-bound free monomers and other structural proteins. Except for the GTP/GDP bound state in the mircotubule all of these factors can be controlled (or altered) by the cell through various mechanisms (some are listed on the 2nd wiki page I linked; I may add some specific ones if I have more time).