- a serious political problem
- a serious, unsolvable problem
The textbook that had led her into this said that there are two types of adjectives: cumulative and coordinate. With coordinate adjectives, you put a comma, but with cumulative ones, you don't.
It's hard to know if this is simply a confusing description of a real situation or whether it is wrong. It suggests that adjectives belong to these two types, which is wrong, but if it really means that adjectives can be listed in two ways, then that's true.
The important point is that it has nothing to do with any particular type of adjective as you can have exactly the same two adjectives with a comma or without. Rather it is a question of coordinated modifiers versus stacked (or what they call cumulative) modifiers. Coordination looks like this: (m , m) h, but stacking looks like this: (m (m h)), where m is a modifier and h is a head noun.
1a) a powerful, new tool (coord)
1b) a powerful new tool (stacked)
In 1a, we have a tool that is both new and powerful. In 1b, we have a new tool that is powerful. This is probably not a distinction worth drawing, but look at this:
2a) a powerful, political force
2b) a powerful political force
In 2a, we have a force that is powerful but also political (as opposed to, say, moral). In 2b) we have a political force that is powerful (as opposed to weak).
3a) a small, happy family
3b) a small happy family
In 3a, the family happens to be both small and happy, but in 3b, the happy family is small (in comparison to most happy families, which we may expect to be big).
Where a modifier is a noun, it is usually indicative of a type and so lends itself to stacking. This also indicates why nouns will come at or towards the end of a string of modifiers.
4a) a big, ugly brick house
But it can also be coordinated:
4b) a big, ugly, brick house
4a is a brick house that is big and ugly. 4b is a house that is big, ugly, and made of brick.