A generic argument that can be used to show that an algebra is non-trivial is [Bergman's Diamond Lemma](http://www.sciencedirect.com/science/article/pii/0001870878900105). In this case, it immediately applies (there are no ambiguities) so for example $x$ is not in the ideal.

This method is useful even in situations in which you cannot find a representation to implement the idea in the other two answers, or when the characteristic gets in the way.