Interrupt Timed Automata (ITA ) form a subclass of stopwatch automata where reachability and some variants of timed model checking are decidable even in presence of parameters. They are well suited to model and analyze real-time operating systems. Here we extend ITA with polynomial guards and updates, leading to the class of polynomial ITA (PolITA ). We prove that reachability is decidable in 2EXPTIME on PolITA, using an adaptation of the cylindrical algebraic decomposition algorithm for the first-order theory of reals. We also obtain decidability for the model checking of a timed version of CTL and for reachability in several extensions of PolITA. In particular, compared to previous approaches, our procedure handles parameters and clocks in a unified way. We also study expressiveness questions for PolITA and show that PolITA are incomparable with stopwatch automata.