Determining U(Z[x]) & U(R[x]) Rings

[joekoviously]

I am having a problem with some abstract algebra and I was wondering if anyone could help and give me some insight. The problem is as follows:

Give an explanation for your answer, long proof not needed:
Determine U(Z[x])
Determine U(R[x])

These are in regards to rings. I know for U(Z[x]) it is something like f(x)=1 g(x)=-1 but I don't know why.

As for U(R[x]) I am rather stuck. Any help or nudging in the right direction would be greatly appreciated.

 

[rasmhop]

For the first case suppose $$f \in \mathbb{Z}[x]$$ is a unit. Then there exists another element $$g \in \mathbb{Z}[x]$$ such that fg=1. If deg f > 0 what can you say about the degree of fg? Can fg be the multiplicative identity when deg fg >0? If deg f = 0 what can we say about f and g?

Similar considerations suffice for the ring of polynomials with real coefficients.