Be careful, "Did you understand step 5?" will get an uniform "Aye!". Check if they understood. Ask for what this step was taken, what alternatives were, why this step was selected, what can be done if this approach fails. Chaining together the various steps is much more important that getting the "rational decomposition of $P(z) / Q(z)$" right (your friendly ...
An interesting example is Sicherman dice: A pair of 6-sided dice, with positive number of eyes on each face that are not the classic 1..6 ones; if you throw them, the distribution of total eyes is the same as for two regular dice.
To construct them, one die is represented by:
&= z + z^2 + z^3 + z^4 + z^5 + z^6 \\